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\title{Trigonometry }

\author{}
\date{}


\begin{document}
\maketitle
In this chapter we will begin studying a very important branch of mathematics called trigonometry, which starts with the ratios of side lengths in right-angled triangles. This topic is of great practical importance to surveyors, architects, engineers and builders, and also has many other applications.

The Greeks had a form of trigonometry based on using lengths of chords of circles. This was further developed by Arab mathematicians. German scholar Regiomontanus (1436-76) wrote the first book on what we might call 'modern' trigonometry to help him in his study of astronomy.

\section*{Introduction}
Suppose we have two similar right-angled triangles as shown on the right. The angles of one match up with the angles of the other, and their matching sides are in the same ratio.

Since the matching sides are in the same ratio, then

\[
\frac{B C}{B^{\prime} C^{\prime}}=\frac{A B}{A^{\prime} B^{\prime}}=\frac{1}{2}
\]

Notice that this can be written as
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-02}

\[
\frac{B C}{A B}=\frac{B^{\prime} C^{\prime}}{A^{\prime} B^{\prime}}=\frac{4}{5}
\]

This tells us that in a right-angled triangle, once the angles are fixed, the ratios of sides are constant.

Trigonometry is concerned with this idea.

\section*{Activity: Similar triangles}
Use your protractor to draw an angle of \(58^{\circ}\). Place markers at distances of \(3 \mathrm{~cm}, 5 \mathrm{~cm}\), and \(8 \mathrm{~cm}\) from \(O\) and draw perpendiculars as shown.

Measure the heights \(h_{1}, h_{2}\) and \(h_{3}\).

You should find that \(h_{1} \approx 4.8 \mathrm{~cm}, h_{2} \approx 8.0 \mathrm{~cm}\) and \(h_{3} \approx 12.8 \mathrm{~cm}\).

Now look at the ratio of the heights to the bases \(\frac{h_{1}}{3}, \frac{h_{2}}{5}\) and \(\frac{h_{3}}{8}\). These ratios are all (approximately) 1.6.

The ratios are the same since the triangles are similar.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-02(2)}
\end{center}

\section*{Naming the sides}
To be more specific about the ratios of sides, we need to introduce some names.

You already know that the longest side in a right-angled triangle, which is the side opposite the right angle, is called the hypotenuse.

We now choose one of the two acute angles and give it a name. One of the Greek letters \(\alpha, \beta, \gamma\) or \(\theta\) is generally used. This angle is sometimes called the reference angle.

The side opposite the reference angle is called the opposite side, often abbreviated to the opposite; the remaining side, which is between the reference angle and the right angle, is called the adjacent side or simply adjacent. The word 'adjacent' means 'near' or 'next to'. The side named the adjacent is next to the

\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-02(1)}
reference angle.

\section*{Example 1}
a Label the three sides of the triangle relative to the marked reference angle.

b Let \(A B=5\) and \(A C=12\).

i Find the length of \(B C\).

ii Write down the ratio \(\frac{\text { opposite }}{\text { hypotenuse }}\).
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-03(1)}

\section*{Solution}
a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-03(2)}
\end{center}

b i By Pythagoras' theorem:

\[
\begin{gathered}
\begin{array}{r}
B C=\sqrt{5^{2}+12^{2}} \\
=13
\end{array} \\
\text { ii } \frac{\text { opposite }}{\text { hypotenuse }}=\frac{12}{13}
\end{gathered}
\]

In this chapter we will generally leave ratios as improper fractions.

For example, we will write \(\frac{12}{5}\) rather than \(2 \frac{2}{5}\).

\section*{Introduction to trigonometry}
\begin{itemize}
  \item For a right-angled triangle, once the angles are fixed, the ratios of sides are constant.
  \item For \(\alpha\), the reference angle, the sides of a right-angled triangle are named as shown on the right
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-03}
\end{center}

\section*{Exercise 13A}
Example 1a 1 For the triangle opposite, name the opposite side, the adjacent side and the hypotenuse, using:

a \(\angle P Q R\) as reference angle

b \(\angle R P Q\) as reference angle

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04(5)}
\end{center}

2 For the triangle opposite, name the opposite side, the adjacent side and the hypotenuse, using:

a \(\angle X Y Z\) as reference angle

b \(\angle X Z Y\) as reference angle

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04(4)}
\end{center}

3 For the triangle opposite, relative to the reference angle \(\alpha\), what is the length of:

a the hypotenuse?

b the opposite side?

c the adjacent side?

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04(7)}
\end{center}

4 For the triangle opposite, relative to the reference angle \(\theta\), what is the length of:

a the hypotenuse?

b the opposite side?

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04(3)}
\end{center}

c the adjacent side?

5 Name the angle that has been used as the reference angle for the naming of the sides.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04}
\end{center}

b \(B\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-04(6)}
\end{center}

d

\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-04(2)}
e \(M\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-05(1)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-05(3)}
\end{center}

6 Draw diagrams of right-angled triangles with:

a reference angle \(30^{\circ}\) and the opposite side of length \(8 \mathrm{~cm}\)

b reference angle \(42^{\circ}\) and the adjacent side of length of \(5 \mathrm{~cm}\)

c reference angle \(40^{\circ}\) and the hypotenuse of length of \(10 \mathrm{~cm}\)

7 Use your protractor to draw an angle of \(31^{\circ}\). Mark off distances of \(4 \mathrm{~cm}, 5 \mathrm{~cm}, 7 \mathrm{~cm}\) and \(10 \mathrm{~cm}\). Draw a perpendicular at each point. Measure the heights and find, approximately, the ratio \(\frac{\text { opposite }}{\text { adjacent }}\) in each triangle.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-05(2)}
\end{center}

\section*{The three basic trigonometric ratios}
We saw in the previous section that once we fix an acute angle in a right-angled triangle, the ratios of the various sides remain the same, irrespective of the size of the triangle. This happens because the triangles are similar.

So in the diagrams on the right:

\[
\frac{A B}{A C}=\frac{A^{\prime} B^{\prime}}{A^{\prime} C^{\prime}} \quad \frac{B C}{A C}=\frac{B^{\prime} C^{\prime}}{A^{\prime} C^{\prime}} \quad \frac{A B}{B C}=\frac{A^{\prime} B^{\prime}}{B^{\prime} C^{\prime}}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-05}
\end{center}

The above ratios have special names based on their relationship to the reference \(\theta\) angle. These names are sine, cosine and tangent respectively, and in the context of their application to \(\theta\), we write:

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-05(4)}
\end{center}

\[
\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }} \quad \cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }} \quad \tan \theta=\frac{\text { opposite }}{\text { adjacent }}
\]

Note: Collectively, the sine, cosine and tangent rations are known as the basic trigonometric ratios. Their names stem from Latin words that reflect a relationship they have with circles.

You will need to learn the three ratios for sine, cosine and tangent carefully. A simple mnemonic is

\section*{SOH CAH TOA}
Sine - Opposite / Hypotenuse

Cosine - Adjacent / Hypotenuse

Tangent - Opposite / Adjacent

\section*{Example 2}
For the triangle shown, write down the value of:
a \(\sin \theta\)
b \(\cos \theta\)
c \(\tan \theta\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-06(1)}
\end{center}

\section*{Solution}
a \(\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}\)

b \(\cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}\)

c \(\tan \theta=\frac{\text { opposite }}{\text { adjacent }}\)

\[
\begin{aligned}
& =\frac{10}{26} \\
& =\frac{5}{13}
\end{aligned}
\]

\[
=\frac{24}{26} \quad=\frac{10}{24}
\]

\(=\frac{12}{13}\)

\(=\frac{5}{12}\)

\section*{Example 3}
Draw the diagonal of a square and find the value of \(\tan 45^{\circ}\).

\section*{Solution}
Two right-angled triangles are formed, each with an angle of \(45^{\circ}\).

Each triangle is isosceles (angle sum of a triangle).

Hence the length of the opposite equals the length of the adjacent, so

\[
\begin{aligned}
\tan 45^{\circ} & =\frac{\text { opposite }}{\text { adjacent }} \\
& =1
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-06}
\end{center}

To find the tangent ratio for other angles, we could draw a triangle and measure lengths. This is what we did in the activity 'Similar triangles' on page 369 when we found the ratio of the height to the base in the triangle containing \(58^{\circ}\). Hence, \(\tan 58^{\circ} \approx 1.6\).

Finding the values of trigonometric ratios in this way can only be approximate. Fortunately, your calculator, using clever mathematics, gives you these values with good accuracy. For example, your calculator gives \(\tan 58^{\circ} \approx 1.600334529\), which is approximately 1.6.

Make sure your calculator is in degree mode. Try entering tan 45 into your calculator and check that you get 1 .

\section*{Example 4}
Use your calculator to find, correct to 3 decimal places:
a \(\tan 15^{\circ}\)
b \(\tan 63^{\circ}\)

Solution

a \(\tan 15^{\circ} \approx 0.268 \quad\) b \(\tan 63^{\circ} \approx 1.963\)

\section*{Values of the trigonometric ratios}
Notice that since the hypotenuse is the longest side in a right-angled triangle, the ratios \(\frac{\text { opposite }}{\text { hypotenuse }}\) and \(\frac{\text { adjacent }}{\text { hypotenuse }}\) are always less than 1 .

That is, if \(0^{\circ}<\theta<90^{\circ}\), then

\[
0<\sin \theta<1 \quad \text { and } \quad 0<\cos \theta<1
\]

\section*{Example 5}
Use a calculator to find, correct to 3 decimal places:
a \(\sin 15^{\circ}\)
b \(\cos 63^{\circ}\)
c \(\sin 23 \frac{1}{2}^{\circ}\)

\section*{Solution}
A calculator gives:
a \(\sin 15^{\circ} \approx 0.259\)
b \(\cos 63^{\circ} \approx 0.454\)
c \(\sin 23 \frac{1}{2}^{\circ} \approx 0.399\)

\section*{Finding the lengths of sides}
We can use sine, cosine and tangent to find the side lengths of right-angled triangles.

Depending on which sides are given, we choose the appropriate ratio: either sine, cosine or tangent. You should always write down the ratio since the unknown side may be in the denominator.

\section*{Example 6}
Find the value of \(x\), correct to 4 decimal places.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-08}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-08(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-08(3)}
\end{center}

\section*{Solution}
a This problem involves opposite and hypotenuse, so use sine.

\[
\begin{aligned}
\frac{x}{8} & =\sin 26^{\circ} \\
x & =8 \times \sin 26^{\circ} \\
& \approx 3.5070
\end{aligned}
\]

b This problem involves adjacent and hypotenuse, so use cosine.

\[
\begin{aligned}
\frac{x}{14} & =\cos 52^{\circ} \\
x & =14 \times \cos 52^{\circ} \\
& \approx 8.6193
\end{aligned}
\]

c This problem involves opposite and adjacent, so use tangent.

\[
\begin{aligned}
\frac{x}{9} & =\tan 52^{\circ} \\
x & =9 \times \tan 52^{\circ} \\
& \approx 11.5195
\end{aligned}
\]

If the unknown is the length of the hypotenuse, an extra step is needed in the calculation.

\section*{Example 7}
Find the value of \(x\), correct to 4 decimal places.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-08(4)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-08(5)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-08(2)}
\end{center}

\section*{Solution}
a This problem involves opposite and hypotenuse, so use sine.

\[
\begin{aligned}
\sin 25^{\circ} & =\frac{12}{x} \\
x \sin 25^{\circ} & =12 \\
x & =\frac{12}{\sin 25^{\circ}} \\
& \approx 28.3944
\end{aligned}
\]

b This problem involves adjacent and hypotenuse, so use cosine.

\[
\begin{aligned}
\cos 16^{\circ} & =\frac{9}{x} \\
x \cos 16^{\circ} & =9 \\
x & =\frac{9}{\cos 16^{\circ}} \\
& \approx 9.3627
\end{aligned}
\]

c This problem involves adjacent and opposite, so use tangent.

\[
\begin{aligned}
& \tan 35^{\circ}=\frac{8}{x} \\
& x \tan 35^{\circ}=8 \\
& x=\frac{8}{\tan 35^{\circ}} \\
& \approx 11.4252
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-09(1)}
\end{center}

\section*{Complementary angles}
In the triangle \(A B C, \sin \theta=\frac{c}{b}\) and \(\cos \left(90^{\circ}-\theta\right)=\frac{c}{b}\). Hence

\[
\sin \theta=\cos \left(90^{\circ}-\theta\right)
\]

Also, \(\cos \theta=\frac{a}{b}\) and \(\sin \left(90^{\circ}-\theta\right)=\frac{a}{b}\). Hence

\[
\cos \theta=\sin \left(90^{\circ}-\theta\right)
\]

For example:

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-09}
\end{center}

\[
\begin{array}{ll}
\sin 70^{\circ}=\cos 20^{\circ} & \cos 40^{\circ}=\sin 50^{\circ} \\
\cos 15^{\circ}=\sin 75^{\circ} & \sin 2^{\circ}=\cos 88^{\circ}
\end{array}
\]

\section*{The three basic trigonometric ratios}
Let \(0^{\circ}<\theta<90^{\circ}\). Then:

\begin{itemize}
  \item \(\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }} \quad \cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }} \quad \tan \theta=\frac{\text { opposite }}{\text { adjacent }}\)
  \item \(\sin \theta=\cos \left(90^{\circ}-\theta\right), \cos \theta=\sin \left(90^{\circ}-\theta\right)\)
  \item \(0<\sin \theta<1 \quad 0<\cos \theta<1 \quad \tan \theta>0\)
\end{itemize}

\section*{Exercise 13B}
1 Write down the value, as a fraction in simplest form, of:

i \(\sin \theta\)

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(3)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(5)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10}
\end{center}

ii \(\cos \theta\)

\(\mathbf{b}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(7)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(6)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(9)}
\end{center}

2 Using your calculator, give the value, to 4 decimal places, of:
a \(\sin 10^{\circ}\)
b \(\sin 20^{\circ}\)
c \(\sin 30^{\circ}\)
d \(\cos 46^{\circ}\)
e \(\cos 75^{\circ}\)
f \(\tan 49^{\circ}\)
g \(\tan 81^{\circ}\)
\(\mathbf{h} \sin 1^{\circ}\)
i \(\cos 37^{\circ}\)
\(\mathbf{j} \cos 88^{\circ}\)
\(\mathbf{k} \tan 3^{\circ}\)
l \(\tan 45^{\circ}\)

3 Find the value of each pronumeral, giving your answer correct to 2 decimal places.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(8)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(1)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-10(2)}
\end{center}

4 Find the value of each pronumeral, giving your answer correct to 3 decimal places.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(5)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(3)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11}
\end{center}

5 Find the value of each pronumeral, giving your answer correct to 4 decimal places.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(6)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(13)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(12)}
\end{center}

6 Find the value of each pronumeral, correct to 4 decimal places.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(9)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(11)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(2)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(10)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(7)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-11(8)}
\end{center}

\(\mathbf{h}\)

\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-11(1)}
i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12(6)}
\end{center}

\(\mathbf{k}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12(2)}
\end{center}

\(\mathbf{j}\)
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-12(8)}

Example 7 7 Find the value of each pronumeral, correct to 4 decimal places.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12(5)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12(1)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12(7)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-12(4)}
\end{center}

\(\mathbf{f}\)

\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-12(3)}
g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13(5)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13}
\end{center}

\(\mathbf{k} k \mathrm{~cm}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13(4)}
\end{center}

\(\mathbf{h}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13(1)}
\end{center}

j

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13(2)}
\end{center}

l

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13(6)}
\end{center}

8 Draw a right-angled triangle with an angle of \(45^{\circ}\). If one of the shorter sides has length 1 , write down the lengths of the other sides without using trigonometry. Hence find the exact values of \(\sin 45^{\circ}, \cos 45^{\circ}\) and \(\tan 45^{\circ}\).

\(9 A B C\) is an equilateral triangle, each of whose sides is 2 units. \(D\) is the midpoint of \(B C\).

a What are the angles \(\angle A B D, \angle B A D\) and \(\angle A D B\) ?

b What is the length of \(B D\) ?

c Find the length of \(A D\) in surd form.

d Use triangle \(A B D\) to complete the following table with exact values (in surd form where necessary).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-13(3)}
\end{center}

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\(\theta\) & \(30^{\circ}\) & \(60^{\circ}\) \\
\hline
\(\sin \theta\) &  &  \\
\hline
\(\cos \theta\) &  &  \\
\hline
\(\tan \theta\) &  &  \\
\hline
\end{tabular}
\end{center}

10 Use complementary angles to complete the following.
a \(\sin 5^{\circ}=\cos \ldots\)
b \(\cos 72^{\circ}=\sin \ldots\)
c \(\sin 65^{\circ}=\cos \ldots\)
d \(\cos 46^{\circ}=\sin \ldots\)

11 Draw a diagram and explain why \(\tan \left(90^{\circ}-\theta\right)=\frac{1}{\tan \theta}\).

In the following questions, \(0^{\circ}<\theta<90^{\circ}\).

12 Find a value of \(\theta\) for which \(\sin \theta=\cos \theta\).

13 a Which is the largest side of a right-angled triangle?

b Is it possible for values of \(\sin \theta\) to be greater than 1 ? Why?

c Is it possible for values of \(\cos \theta\) to be greater than 1 ? Why?

d Is it possible for values of \(\tan \theta\) to be greater than 1 ? Why?

14 a What happens to the value of \(\sin \theta\) as \(\theta\) gets closer and closer to \(0^{\circ}\) ?

b What happens to the value of \(\sin \theta\) as \(\theta\) gets closer and closer to \(90^{\circ}\) ?

c Can you explain why the value of \(\sin \theta\) behaves like this?

15 Repeat Question 14 for:
i \(\cos \theta\)
ii \(\tan \theta\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-14}
\end{center}

\section*{Finding angles}
We have seen that in any right-angled triangle with reference angle \(\theta\), there are three basic ratios associated with that angle and that the value of each ratio can be found using the calculator. In order to use the trigonometric ratios to find angles in a right-angled triangle, we need to reverse the process. Thus, given the sine, cosine or tangent of some angle between \(0^{\circ}\) and \(90^{\circ}\), we want to find the angle with the given ratio.

We have seen that \(\tan 45^{\circ}=1\). We say that \(45^{\circ}\) is the inverse tangent of 1 .

This is written as \(\tan ^{-1} 1=45^{\circ}\)

We use the calculator to find \(\tan ^{-1} x\). For example, \(\tan 37^{\circ}=0.735 \ldots\) and \(\tan ^{-1} 0.735\) is approximately equal to \(37^{\circ}\), correct to 2 decimal places.

Similarly, \(\sin 30^{\circ}=\frac{1}{2}\). We say \(30^{\circ}\) is the inverse sine of \(\frac{1}{2}\) and write \(\sin ^{-1} 0.5=30^{\circ}\)

This notation is standard, but can be rather misleading. The index -1 does NOT mean 'one over' as it normally does in algebra. To help you avoid confusion, you should always read \(\sin ^{-1} x\) as inverse sine of \(x\) and \(\tan ^{-1} x\) as inverse tan of \(x\) and so on.

Similarly, a calculator gives \(\cos 35^{\circ} \approx 0.8192\) and \(\cos ^{-1} 0.8192 \approx 35^{\circ}\) (read this as inverse cosine of 0.8192 ).

\section*{Finding angles in right-angled triangles}
\begin{itemize}
  \item If \(x>0\), then \(\tan ^{-1} x\) is the angle whose tangent is \(x\).
  \item If \(0<x<1\), then \(\sin ^{-1} x\) is the angle whose sine is \(x\).
  \item If \(0<x<1\), then \(\cos ^{-1} x\) is the angle whose cosine is \(x\).
  \item Tangent, sine and cosine are applied to a reference angle to determine a specific ratio of side lengths in a right-angled triangle.
  \item Inverse tangent, inverse sine and inverse cosine are applied to a ratio to determine the size of a specific reference angle.
\end{itemize}

\section*{Example 8}
Find, correct to the nearest degree:
a \(\sin ^{-1} 0.6\)
b \(\cos ^{-1} 0.412\)
c \(\tan ^{-1} 2\)
d the angle \(\theta\) for which \(\sin \theta=0.8\)
e the angle \(\theta\) for which \(\cos \theta=0.2\)
f \(\cos ^{-1} 2\)

\section*{Solution}
a By calculator

\[
\begin{aligned}
\sin ^{-1} 0.6 & =36.86989 \ldots \\
& \approx 37^{\circ}
\end{aligned}
\]

c \(\tan ^{-1} 2=63.43494 \ldots \circ\)

\[
\approx 63^{\circ}
\]

b \(\cos ^{-1} 0.412=65.66946 \ldots{ }^{\circ}\)

\(\approx 66^{\circ}\)

d if \(\sin \theta=0.8\)

then \(\quad \theta=\sin ^{-1} 0.8\)

\(=53.13 \ldots \circ\)

\(\approx 53^{\circ}\)

e If \(\cos \theta=0.2\)

f \(2>1\), so \(\cos ^{-1} 2\) is not defined.

\[
\begin{aligned}
& =78.463 \ldots \circ \\
& \approx 78^{\circ}
\end{aligned}
\]

\section*{Example 9}
Find each angle, correct to the nearest degree.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-15(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-15}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-15(2)}
\end{center}

\section*{Solution}
a \(\sin \theta=\frac{8}{12}=\frac{2}{3}\)

b \(\cos \beta=\frac{9}{16}\)

c \(\tan \alpha=\frac{12}{5}\)

so \(\theta=\sin ^{-1} \frac{2}{3}\)

so \(\beta=\cos ^{-1} \frac{9}{16}\)

so \(\alpha=\tan ^{-1} \frac{12}{5}\)

\(\approx 42^{\circ}\)

\(\approx 56^{\circ}\)

\(\approx 67^{\circ}\)

\section*{Exercise 13C}
1 Find, correct to the nearest degree:
a \(\sin ^{-1} 0.7\)
b \(\sin ^{-1} 0.732\)
c \(\cos ^{-1} 0.713\)
d \(\cos ^{-1} 0.1234\)
e \(\tan ^{-1} 0.1\)
f \(\tan ^{-1} 12\)

2 Find \(\theta\), correct to the nearest degree.
a \(\sin \theta=0.4067\)
b \(\sin \theta=0.8480\)
c \(\cos \theta=0.9816\)
d \(\cos \theta=0.5299\)
e \(\tan \theta=1.5399\)
f \(\tan \theta=4.0108\)

3 Find the size of each marked angle, correct to the nearest degree.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(7)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(2)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(6)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(4)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(5)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(1)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-16(3)}
\end{center}

4 What are the sizes of the angles, correct to the nearest degree, in a:

a \(3,4,5\) right-angled triangle?

b \(5,12,13\) right-angled triangle?

c \(7,24,25\) right-angled triangle?

5 Find the angle between the diagonal and the base of the rectangle, correct to 2 decimal places.

6 Find the base angle in the isosceles triangle, correct to

\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-17(1)}
2 decimal places.

7 The diagonals of a rhombus have lengths \(7 \mathrm{~cm}\) and \(10 \mathrm{~cm}\). Find the angles of the rhombus correct to 2 decimal places.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-17}
\end{center}

\section*{Miscellaneous exercises}
The following exercise reinforces the techniques introduced in the previous sections.

You will have found the exact values of sine, cosine and tangent for \(30^{\circ}, 45^{\circ}\) and \(60^{\circ}\) in previous exercises. These values are exemplified in the following triangles. One is a right-angled isosceles triangle with shorter sides of 1 unit, and the other is an equilateral triangle with side lengths 2 and with an angle bisector drawn.
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-17(2)}

These trigonometric ratios are given in the table below.

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\(\theta\) & \(\sin \theta\) & \(\cos \theta\) & \(\tan \theta\) \\
\hline
\(30^{\circ}\) & \(\frac{1}{2}\) & \(\frac{\sqrt{3}}{2}\) & \(\frac{1}{\sqrt{3}}\) \\
\hline
\(45^{\circ}\) & \(\frac{1}{\sqrt{2}}\) & \(\frac{1}{\sqrt{2}}\) & 1 \\
\hline
\(60^{\circ}\) & \(\frac{\sqrt{3}}{2}\) & \(\frac{1}{2}\) & \(\sqrt{3}\) \\
\hline
\end{tabular}
\end{center}

Check the details in the triangles and the entries in the table. We can either learn the table or remember the diagrams to construct the table. Note that, by complementary angles, \(\sin 30^{\circ}=\cos 60^{\circ}, \sin 60^{\circ}=\cos 30^{\circ}\) and \(\sin 45^{\circ}=\cos 45^{\circ}\). Knowing that \(\cos 60^{\circ}=\frac{1}{2}\) and \(\tan 45^{\circ}=1\), we can quickly reconstruct the triangles to find all the
values.

\section*{Exercise 13D}
1 Find the value of the pronumerals, giving angles to the nearest degree and lengths correct to 2 decimal places.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(9)}
\end{center}

g
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-18(6)}

\(\mathbf{m}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(11)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(2)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(10)}
\end{center}

k

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(13)}
\end{center}

n

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(5)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18}
\end{center}

15

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(8)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(7)}
\end{center}

l

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(12)}
\end{center}

o

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-18(1)}
\end{center}

2 Find the exact value of the pronumerals.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(6)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(7)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(4)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(8)}
\end{center}

\(\mathbf{h}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(5)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(2)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19(1)}
\end{center}

i

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-19}
\end{center}

\section*{Solving problems using trigonometry}
The results of the previous sections can be applied to practical problems.

In tackling each problem you should draw a diagram and clearly identify the side or angle you are asked to find.

\section*{Example 10}
A plane climbs at an angle of \(4^{\circ}\) after take-off. Find its altitude when it has flown a horizontal distance of \(500 \mathrm{~m}\).

\section*{Solution}
Let the altitude be \(x \mathrm{~m}\).

Then \(\tan 4^{\circ}=\frac{x}{500}\)

\[
\begin{aligned}
x & =500 \tan 4^{\circ} \\
& \approx 35
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-20}
\end{center}

Thus, the plane's altitude is \(35 \mathrm{~m}\) (to the nearest metre).

\section*{Example 11}
A rope, \(4 \mathrm{~m}\) long, is attached to a vertical pole. The rope, held taut, is pegged into the ground \(2 \mathrm{~m}\) from the base of the pole.

a Find the angle the rope makes with the ground.

b Find the angle the rope makes with the pole.

\section*{Solution}
a Let \(P Q\) be the pole and \(\theta\) be the angle between the rope and the ground.

Then \(\cos \theta=\frac{2}{4}\)

\[
=\frac{1}{2}
\]

Hence \(\quad \theta=\cos ^{-1} \frac{1}{2}\)

\[
=60^{\circ}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-20(1)}
\end{center}

Thus, the rope makes an angle of \(60^{\circ}\) with the ground.

b \(\angle \mathrm{QPR}=90^{\circ}-60^{\circ}\)

\[
=30^{\circ}
\]

Thus, the rope makes an angle of \(30^{\circ}\) with the pole.

\section*{Example 12}
A shop sign is supported by two rods. One rod is horizontal and is of length \(2 \mathrm{~m}\). The other rod is of length \(x \mathrm{~m}\) and makes an angle of \(36^{\circ}\) with the horizontal. Find its length correct to 2 decimal places.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-21(1)}
\end{center}

\section*{Solution}
From the diagram \(\cos 36^{\circ}=\frac{2}{x}\)

\[
x \cos 36^{\circ}=2
\]

Hence

\[
\begin{aligned}
x & =\frac{2}{\cos 36^{\circ}} \\
& \approx 2.47
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-21(2)}
\end{center}

Therefore the rod is of length \(2.47 \mathrm{~m}\) (correct to 2 decimal places).

\section*{Angles of elevation and depression}
When a person looks at an object that is higher than the person, the angle between the line of sight and the horizontal is called the angle of elevation.

On the other hand, when the object is lower than the person, the angle between the horizontal and the line of sight is called the angle of depression.

Angles of elevation and angles of depression are always measured from the horizontal.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-21(3)}
\end{center}

\section*{Example 13}
From a point \(P, 30 \mathrm{~m}\) from a building, the angle of elevation of the top of the building is \(41^{\circ}\). Find the height of the building, correct to the nearest metre.

\section*{Solution}
Let \(h \mathrm{~m}\) be the height of the building.

Now \(\tan 41^{\circ}=\frac{h}{30}\)

hence

\[
\begin{aligned}
\mathrm{h} & =30 \tan 41^{\circ} \\
& \approx 26
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-21}
\end{center}

So the building is \(26 \mathrm{~m}\) high (correct to the nearest metre).

\section*{Example 14}
From the top of a cliff, \(100 \mathrm{~m}\) above sea level, the angle of depression to a ship sailing below is \(17^{\circ}\). How far is the ship from the base of the cliff, to the nearest metre?

\section*{Solution}
The diagram to the right shows the top of the cliff \(P\), the ship \(S\) and the base of the cliff \(B\). Let \(S B=x \mathrm{~m}\) be the distance of the ship from the cliff. By alternate angles, \(\angle P S B=17^{\circ}\).

Hence \(\tan 17^{\circ}=\frac{100}{x}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-22(1)}
\end{center}

so \(\quad \begin{aligned} x & =\frac{100}{\tan 17^{\circ}} \\ & =327.085\end{aligned}\)

The distance is \(327 \mathrm{~m}\) (correct to the nearest metre).

\section*{Compound problems}
\section*{Example 15}
From a point \(A, 10 \mathrm{~m}\) from the base of a tree of height \(h \mathrm{~m}\), the angle of elevation of the top of a tree is \(36^{\circ}\). From a point \(B, x \mathrm{~m}\) further away from the base of the tree, the angle of elevation is \(20^{\circ}\).

a Draw a diagram to illustrate the information given.

b Find the height of the tree, correct to the nearest tenth of a metre.

c Find the distance \(x \mathrm{~m}\), correct to the nearest tenth of a metre.

\section*{Solution}
a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-22}
\end{center}

b \(\quad\) In \(\triangle O H A \quad \tan 36^{\circ}=\frac{h}{10}\)

so

\[
\begin{aligned}
h & =10 \tan 36^{\circ} \\
& =7.2654 \ldots \text { (leave this answer in your calculator) } \\
& \approx 7.3 \text { (correct to the nearest tenth of a metre) }
\end{aligned}
\]

(Continued over page)
c In \(\triangle H O B \quad \frac{h}{O B}=\tan 20^{\circ}\)

Hence \(\quad O B=\frac{h}{\tan 20^{\circ}}\)

\[
=\frac{10 \tan 36^{\circ}}{\tan 20^{\circ}}
\]

\[
=19.96 \ldots
\]

so

\[
x=A B
\]

\(=O B-O A\)

\(=9.96\)

\(\approx 10.0\) (correct to the nearest tenth of a metre)

Note: Do not re-enter approximations. Keep values in your calculator.

\section*{Angles of elevation and depression}
\begin{itemize}
  \item When the object is higher than the person, the angle between the line of sight and the horizontal is called the angle of elevation.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-23(2)}
\end{center}

\begin{itemize}
  \item When the object is lower than the person, the angle between the line of sight and the horizontal is called the observer angle of depression.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-23}
\end{center}

\section*{Exercise 13E}
1 A \(5 \mathrm{~m}\) ladder leans against a wall, making an angle of \(50^{\circ}\) with the floor. How far up the wall (correct to the nearest centimetre) is the top of the ladder?

2 A girl's shadow is \(60 \mathrm{~cm}\) long. If the line joining the top of the girl's head and the end of the shadow makes an angle of \(68^{\circ}\) with the ground, how tall is the girl (correct to the nearest centimetre)?
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-23(1)}

3 Gregor drives up a hill that is inclined at an angle of \(15^{\circ}\) to the horizontal. If he drives for \(5 \mathrm{~km}\), through what vertical distance has he risen? Give your answer correct to the nearest metre.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-24}
\end{center}

4 A tower is held in place by a rope of length \(25 \mathrm{~m}\). The rope is attached to the top of the tower and makes an angle of \(76^{\circ}\) with the ground. Find:

a the height of the tower, correct to the nearest centimetre

b how far, to the nearest centimetre, the base of the tower is from the anchor point

5 A ladder of length \(4 \mathrm{~m}\) is leaning against a wall so that the top of the ladder is at the top of the wall. If the ladder makes an angle of \(7^{\circ}\) with the wall, find, correct to the nearest centimetre:

a the height of the wall

b how far the base of the ladder is from the wall

6 A skateboard ramp is \(5 \mathrm{~m}\) long and rises a vertical distance of \(2 \mathrm{~m}\). What angle does the ramp make with the horizontal ground? Give your answer correct to the nearest degree.

7 An isosceles triangle has base length \(10 \mathrm{~cm}\) and the other two sides are each of length \(6 \mathrm{~cm}\). Find the angles in the triangle, giving your answers correct to the nearest degree.

8 A roof truss is \(6.8 \mathrm{~m}\) wide and \(1.5 \mathrm{~m}\) high. Calculate the angle \(\theta\) between the two beams, correct to the nearest degree.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-24(1)}
\end{center}

9 A horizontal awning that has a depth of \(80 \mathrm{~cm}\) is attached to the top of a window of height \(2 \mathrm{~m}\). What is the least angle that a ray of sunlight can make with the window if no light is to shine directly into it? Give your answer correct to the nearest tenth of a degree.

10 The angle of elevation from a point on a horizontal surface to the top of a vertical pole \(20 \mathrm{~m}\) away is \(35^{\circ}\). Find the height of the pole, correct to the nearest metre.

11 From a point on top of a tower of height \(5.8 \mathrm{~m}\), the angle of depression to a swimmer is \(16^{\circ}\). How far from the base of the tower is the swimmer? Give your answer correct to the nearest tenth of a metre.

12 Two points in space \(25 \mathrm{~m}\) apart are such that the vertical distance between them is \(18 \mathrm{~m}\). Find the angle of elevation or depression between the two points. Give your answer correct to the nearest degree.

13 Two bushwalkers, Gore and Tess, are testing new electronic measuring devices. Gore is on a mountain side at the \(2300 \mathrm{~m}\) contour line. Tess is at the \(2150 \mathrm{~m}\) contour line on a different mountain. Their measuring devices say that the direct distance between them is \(1.35 \mathrm{~km}\). Find the angle of elevation, correct to 2 decimal places, from Tess to Gore.

14 Carlos is standing \(16 \mathrm{~m}\) from the base of a tree that is \(14 \mathrm{~m}\) high. Carlos's eye is \(1.6 \mathrm{~m}\) from the ground.

a What is the angle of elevation from Carlos's eye to the top of the tree, correct to the nearest degree?
b If Carlos looks up at an angle of \(26^{\circ}\), how far up the tree will he see? Give your answer correct to the nearest centimetre.

c What is the angle of elevation from Carlos's eye to a point halfway up the tree, correct to the nearest degree?

15 From a point \(100 \mathrm{~m}\) from the base of a building, the angle of elevation to the top of the building is \(31^{\circ}\).

a How high is the building, correct to the nearest tenth of a metre?

b If there is a vertical flag pole of height \(15 \mathrm{~m}\) on top of the building, what is the angle of elevation of the top of the flag pole? Give your answer correct to the nearest degree.

16 For the diagram on the right:

a find the value of \(h\), correct to 2 decimal places

b use this value to find \(x\), correct to 2 decimal places

17 For the diagram on the right:

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-25}
\end{center}

a find the value of \(h\), correct to the nearest centimetre

b use this value to find the length \(A C\), correct to the nearest centimetre

c hence find \(x\), correct to the nearest centimetre

18 For the diagram below:
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-25(1)}

a find the value of \(h\), correct to the nearest centimetre

b hence find the value of \(\theta\), correct to the nearest degree

19 The top of a \(25 \mathrm{~m}\) tower is observed from two points, \(A\) and \(B\), in line with the base of the tower, but on opposite sides of it. The angle of elevation of the top of the tower is \(26^{\circ}\) from \(A\) and \(34^{\circ}\) from \(B\).

a Draw a diagram to represent the information above.

b How far apart are the points \(A\) and \(B\), correct to the nearest centimetre?

20 At a point \(A\) on the ground, \(12 \mathrm{~m}\) from the base of a building, the angle of elevation of the top of the building is \(51^{\circ}\). From a point \(B\) on the ground \(x \mathrm{~m}\) further out from \(A\), in line with the base of the building, the angle of elevation is \(40^{\circ}\).

a Draw a diagram to represent the information above.

b Find the height of the building to the nearest centimetre.

c Find the value of \(x\) correct to the nearest centimetre.

21 The line \(y=3 x+6\) crosses the \(x\)-axis at an angle of \(\theta\) with the positive direction of the \(x\)-axis. Find the value of \(\theta\), correct to 1 decimal place.

\section*{True bearings}
Bearings are used to indicate the direction of an object from a fixed reference point. True bearings give the angle \(\theta^{\circ}\) from the north, measured clockwise. We write a true bearing of \(\theta^{\circ}\) as \(\theta^{\circ} \mathrm{T}\), where \(\theta^{\circ}\) is an angle between \(0^{\circ}\) and \(360^{\circ}\). It is customary to write the angle using three digits, so \(0^{\circ} \mathrm{T}\) is written \(000^{\circ} \mathrm{T}, 15^{\circ} \mathrm{T}\) is written \(015^{\circ} \mathrm{T}\) and so on.

For example, the true bearing of \(A\) from \(O\) is \(060^{\circ} \mathrm{T}\), and the true bearing of \(B\) from \(O\) is \(140^{\circ} \mathrm{T}\).

The true bearing of \(C\) from \(O\) is \(350^{\circ} \mathrm{T}\).
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-26(1)}

\section*{Example 16}
Write down the true bearings of \(A, B\) and \(C\) from \(O\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-26}
\end{center}

\section*{Solution}
The true bearing of \(A\) from \(O\) is \(030^{\circ} \mathrm{T}\).

For \(B, 30^{\circ}+180^{\circ}=210^{\circ}\), so the true bearing of \(B\) from \(O\) is \(210^{\circ} \mathrm{T}\).

For \(C, 360^{\circ}-20^{\circ}=340^{\circ}\), so the true bearing of \(C\) from \(O\) is \(340^{\circ} \mathrm{T}\).

\section*{Example 17}
Anthony walks for \(490 \mathrm{~m}\) on a true bearing of \(140^{\circ} \mathrm{T}\) from point \(A\) to point \(B\).

Find, correct to the nearest metre:

a how far east the point \(B\) is from point \(A\)

b how far south the point \(B\) is from point \(A\)

\section*{Solution}
a Distance east \(=490 \sin 40^{\circ}\)

\[
\begin{aligned}
& =314.96 \ldots \\
& \approx 315 \mathrm{~m}
\end{aligned}
\]

b Distance south \(=490 \cos 40^{\circ}\)

\[
\begin{aligned}
& =375.36 \ldots \\
& \approx 375 \mathrm{~m}
\end{aligned}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-27(1)}
\end{center}

\section*{Example 18}
The bearing of \(B\) from \(A\) is \(075^{\circ}\). Find the bearing of \(A\) from \(B\).

\section*{Solution}
Mark the direction of \(B\) from \(A\) with respect to the direction of north, then re-draw a compass cross with origin at \(B\).

\(\angle A B Y=75^{\circ}\) (alternate)

\(\angle A B X=15^{\circ}\) (sum of angles in triangle \(A B X\) )

\(180^{\circ}+75^{\circ}=255^{\circ}\left(\right.\) or \(\left.270^{\circ}-15^{\circ}=255^{\circ}\right)\)

So the true bearing of \(A\) from \(B\) is \(255^{\circ}\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-27}
\end{center}

\section*{Exercise 13F}
1 For the diagram opposite, write down the true bearing of points \(A, B, C\) and \(D\) from \(O\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-27(2)}
\end{center}

2 Draw a diagram to illustrate the fact that:

a the bearing from \(O\) to \(Z\) is \(160^{\circ} \mathrm{T}\)

b the bearing from \(P\) to \(Q\) is \(340^{\circ} \mathrm{T}\)

c to get from \(C\) to \(D\), you travel on a bearing of \(210^{\circ} \mathrm{T}\)

d to get from \(A\) to \(Y\), you travel on a bearing of \(170^{\circ} \mathrm{T}\)

3 Using a protractor, measure the bearing of:
a \(A\) from \(C\)
b \(B\) from \(C\)

4 Jo hikes for \(369 \mathrm{~m}\) on a bearing of \(053^{\circ} \mathrm{T}\) from point \(A\) to point \(B\). Find, correct to the nearest metre:

a how far east point \(B\) is from point \(A\)

b how far north point \(B\) is from point \(A\)

5 A ship leaves port and sails on a bearing of \(160^{\circ} \mathrm{T}\) for \(200 \mathrm{~km}\). Find, correct to the nearest tenth of a kilometre:

a how far south the ship is from the port

b how far east the ship is from the port

6 During an orienteering competition, Fran runs \(1600 \mathrm{~m}\) on a bearing of \(316^{\circ} \mathrm{T}\) from checkpoint 1 to checkpoint 2.

a How far west of checkpoint 1 is checkpoint 2, correct to the nearest metre?

b How far north of checkpoint 1 is checkpoint 2, correct to the nearest metre?

7 Alan is standing \(100 \mathrm{~m}\) due east of a landmark. Benny is standing \(250 \mathrm{~m}\) due south of the landmark. What is the bearing, correct to the nearest tenth of a degree, of:

a Benny from Alan?

b Alan from Benny?

8 Christos is \(200 \mathrm{~m}\) from a tree and Danielle is \(400 \mathrm{~m}\) from the same tree. What is the bearing of Danielle from Christos, to the nearest degree, if:

a Christos is due north of the tree and Danielle is due east of the tree?

b Christos is due west of the tree and Danielle is due south of the tree?

c Christos is due east of the tree and Danielle is due south of the tree?

9 A hiker starts at point \(O\) and travels for \(6 \mathrm{~km}\) on a bearing of \(070^{\circ} \mathrm{T}\) to a point \(A\). The hiker wishes to travel from \(A\) to a point \(B\), which is \(15 \mathrm{~km}\) due east of \(O\). Give answers correct to the nearest hundredth of a kilometre and to the nearest degree.
a How far east of \(O\) is \(A\) ?
b How far north of \(O\) is \(A\) ?
c How far east of \(A\) is \(B\) ?
d How far south of \(A\) is \(B\) ?
e How far is \(B\) from \(A\) ?
f On what bearing does the hiker travel if she walks directly from \(A\) to \(B\) ?

10 A car travels on a bearing of \(280^{\circ} \mathrm{T}\) on a very long straight road at \(85 \mathrm{~km} / \mathrm{h}\), starting from an initial point at 12 p.m. Find how far north the car is from its original point at 12:30 p.m., correct to one-tenth of a kilometre.

\section*{Review exercise}
1 Write down the sine, cosine and tangent ratio of the indicated angle. Leave answers in exact form.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(2)}
\end{center}

2 Find the value of each pronumeral, correct to 2 decimal places.

a
\includegraphics[max width=\textwidth, center]{2023_12_09_b4f6b04009fdb58c04c0g-29(9)}

e

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(6)}
\end{center}

g

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(3)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(5)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(7)}
\end{center}

\(\mathbf{f}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(8)}
\end{center}

h

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-29(4)}
\end{center}

3 Find the value of each pronumeral, correct to the nearest degree.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-30(1)}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-30}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-30(3)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-30(2)}
\end{center}

4 A ladder \(9 \mathrm{~m}\) long is resting against a wall, making an angle of \(55^{\circ}\) with the ground.

a How high up the wall does the ladder reach, correct to the nearest centimetre?

b How far away from the wall is the foot of the ladder, correct to the nearest centimetre?

5 A ship is travelling due north at \(15 \mathrm{~km} / \mathrm{h}\). The navigator sees a light due east, and eight minutes later the light has a bearing of \(125^{\circ} \mathrm{T}\) from the ship. What is the distance from the ship to the light each time the navigator sees it, correct to the nearest metre?

6 The sides of a rectangle are \(4 \mathrm{~cm}\) and \(5 \mathrm{~cm}\) in length.

a Find the angles that a diagonal makes with the sides, correct to the nearest degree.

b What are the angles of intersection of the diagonals, correct to the nearest degree?

7 The angle of elevation of the top of a vertical cliff is observed to be \(41^{\circ}\) from a boat \(450 \mathrm{~m}\) from the base of the cliff. What is the height of the cliff, correct to the nearest metre?

8 Find the angle of depression, to the nearest degree, of a boat \(500 \mathrm{~m}\) out to sea as observed from the top of a cliff of height \(90 \mathrm{~m}\).

9 A ship sails on a bearing of \(156^{\circ} \mathrm{T}\) until it is \(45 \mathrm{~km}\) south of its starting point. How far has it travelled, correct to 2 decimal places?

10

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-30(4)}
\end{center}

a Find the height \(h\), correct to the nearest centimetre.

b Find the value of \(x\), correct to the nearest centimetre.

\section*{Challenge exercise}
1 The 'good ship' is at \(A, 1 \mathrm{~km}\) due north of the 'enemy ship' at 12 p.m. After this time the good ship moves on a bearing of \(045^{\circ} \mathrm{T}\) while the enemy ship remains stationary.

a The good ship has travelled \(1.5 \mathrm{~km}\) from \(A\). Find, to the nearest metre:

i how far east it is from its position at 12 p.m.

ii how far north it is from its position at 12 p.m.

iii how far it is from the enemy ship

b Suppose that the good ship travels from \(A\) on the same bearing for \(x \mathrm{~km}\). Find the distance between the ships \(d \mathrm{~km}\), in terms of \(x\).

c Finally, suppose that the good ship travels from \(A\) at a speed of \(10 \mathrm{~km} / \mathrm{h}\). How far apart are the ships at 2:45 p.m. that day, correct to 2 decimal places?

2 A police helicopter, hovering at an altitude of \(1000 \mathrm{~m}\), observes a car travelling along a straight highway. At that instant, the angle of depression of the car from the helicopter is \(27^{\circ}\). Five seconds later, the angle of depression of the car from the helicopter is \(26^{\circ}\).

a What is the horizontal distance the car is from the helicopter when it is first sighted, giving your answer correct to the nearest tenth of a metre?

b Calculate the horizontal distance the car is from the helicopter 5 seconds after it is first sighted, giving your answer correct to the nearest tenth of a metre.

c Find the distance the car has travelled in the 5 seconds, giving your answer correct to the nearest tenth of metre.

d What is the speed at which the car is travelling in metres per second, correct to 1 decimal place?

3 Consider the diagram shown below.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-31}
\end{center}

a Prove that \(B D=\frac{1}{2}(A D+C D)\).

b Hence, deduce that \(\tan D E B=\frac{1}{2} \tan \angle A E D+\frac{1}{2} \tan \angle C E D\).

4 Consider the diagram shown below.

a Write down the values of \(\sin \theta\) and \(\cos \theta\).

b Show that \((\sin \theta)^{2}+(\cos \theta)^{2}=1\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-32}
\end{center}

5 a Find the area in square metres of the triangle shown below, correct to 2 decimal places.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-32(3)}
\end{center}

b Prove that the area of triangle \(A B C\) is \(\frac{1}{2} a b \sin \theta\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-32(1)}
\end{center}

6 a For the diagram shown below, find the value of \(d\).

b Show that \(x=10(\sqrt{3}-1)\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_b4f6b04009fdb58c04c0g-32(2)}
\end{center}


\end{document}