Right-triangle trigonometry

Warm-up

Three quick questions from earlier lessons.

Discover

A ladder 10 m long leans against a wall and makes a $60^\circ$ angle with the ground. How high up the wall does it reach?

$10\sin 60^\circ \approx 8.7$ m
$10\cos 60^\circ = 5$ m

Key idea

In a right triangle, each acute angle gives three ratios of sides.

\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}

,

\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}

,

\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}

(SOH CAH TOA). To find an angle from two sides, use

\sin^{-1}

,

\cos^{-1}

or

\tan^{-1}

.

Worked example

A ramp rises 2 m over a horizontal distance of 7 m. Find its angle with the ground.

\tan\theta = \dfrac{2}{7}

2 m is opposite the angle; 7 m is adjacent.

\theta = \tan^{-1}\dfrac{2}{7} \approx 15.9^\circ

Undo tan with tan⁻¹ (calculator in degrees).

Your turn, guided

Your turn: the hypotenuse is 8 and the angle is $30^\circ$ . Find the side opposite it.

\text{opposite} = 8\sin 30^\circ

SOH.

= 4

sin 30° = 1/2.

Practice ★

A right triangle has hypotenuse 15 and an angle of $30^\circ$ . How long is the side opposite this angle? Give the exact value.
A right triangle has hypotenuse 17 and an angle of $30^\circ$ . How long is the side opposite this angle? Give the exact value.
A right triangle has hypotenuse 11 and an angle of $30^\circ$ . How long is the side opposite this angle? Give the exact value.
A right triangle has hypotenuse 12 and an angle of $30^\circ$ . How long is the side opposite this angle? Give the exact value.

Watch out!

Practice ★★

You stand 34 m from the foot of a tree and see its top at an angle of elevation of $40^\circ$ . How tall is the tree, to the nearest tenth of a meter? (Ignore your own height.)
You stand 26 m from the foot of a tree and see its top at an angle of elevation of $20^\circ$ . How tall is the tree, to the nearest tenth of a meter? (Ignore your own height.)
You stand 24 m from the foot of a tree and see its top at an angle of elevation of $50^\circ$ . How tall is the tree, to the nearest tenth of a meter? (Ignore your own height.)

In a right triangle, the side opposite angle $\theta$ is $5$ and the hypotenuse is $10$ . Find $\theta$ in degrees.

In a right triangle, the side opposite angle $\theta$ is 8 and the hypotenuse is 11. Find $\theta$ in degrees, to 1 decimal place.
In a right triangle, the side opposite angle $\theta$ is 6 and the hypotenuse is 15. Find $\theta$ in degrees, to 1 decimal place.

Exit ticket

Three questions to check you have got it.

You stand 34 m from the foot of a tree and see its top at an angle of elevation of $40^\circ$ . How tall is the tree, to the nearest tenth of a meter? (Ignore your own height.)
You stand 26 m from the foot of a tree and see its top at an angle of elevation of $20^\circ$ . How tall is the tree, to the nearest tenth of a meter? (Ignore your own height.)
You stand 24 m from the foot of a tree and see its top at an angle of elevation of $50^\circ$ . How tall is the tree, to the nearest tenth of a meter? (Ignore your own height.)

Summary card

Right-triangle trigonometry

SOH CAH TOA.
Side from an angle: multiply ( $\text{opposite} = \text{hyp} \cdot \sin\theta$ ).
Angle from two sides: $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ .

Watch out: name the sides from the angle you use.

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