Trigonometry

Spec reference: Geometry and measures - Pythagoras' theorem and trigonometry
Key idea: Use sine, cosine and tangent to find missing sides and angles in right-angled triangles.

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SOH CAH TOA

In a right-angled triangle, for an angle $\theta$:

$$\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$$

• Hypotenuse - longest side (opposite right angle)
• Opposite - side opposite the angle $\theta$
• Adjacent - side next to the angle $\theta$ (not the hypotenuse)

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Finding a missing side

Example

Find side $x$ in a triangle where the hypotenuse is 12 cm and angle $\theta =35°$.

The side $x$ is adjacent to $\theta$.

$$\cos 35°=\frac{x}{12}$$$$x=12\cos 35°=12\times 0.819=\boxed{{\displaystyle 9.83\text{ cm}}}$$

Example

Find the opposite side when the adjacent is 9 cm and $\theta =50°$.

$$\tan 50°=\frac{\text{opp}}{9}$$$$\text{opp}=9\tan 50°=9\times 1.192=\boxed{{\displaystyle 10.7\text{ cm}}}$$

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Finding a missing angle

Use the inverse trigonometric functions: ${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$.

Example

Find angle $\theta$ if the opposite is 7 cm and hypotenuse is 10 cm.

$$\sin \theta =\frac{7}{10}=0.7$$$$\theta ={\sin }^{-1}(0.7)=\boxed{{\displaystyle 44.4°}}$$

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Exact trigonometric values

Angle	$\sin$	$\cos$	$\tan$
$0°$	$0$	$1$	$0$
$30°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45°$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90°$	$1$	$0$	undefined

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Exam tips

Watch out for

• Make sure your calculator is in degree mode (not radians)
• Always label sides relative to the angle you are working with
• To find an angle, use inverse trig: ${\sin }^{-1}$, not $\frac{1}{\sin }$

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Test Yourself

Question 1 of 5

$\sin\theta = ?$