Trigonometric ratios in right triangles

Warm-up

Three quick questions from earlier lessons.

Discover

Two right triangles of different sizes both have a $35^\circ$ angle. Is the ratio opposite/hypotenuse the same in both?

Yes
No, the bigger triangle has a bigger ratio

Make the triangle bigger and smaller. Divide the opposite side by the hypotenuse.

It stays 0.6
It grows with the triangle

Key idea

Ratios that depend only on the angle.

For an acute angle

A

of a right triangle:

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

,

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

,

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

(SOH-CAH-TOA).

Worked example

Legs 8 (opposite $A$ ) and 15, hypotenuse 17.

\sin A = \tfrac{8}{17}

SOH.

\cos A = \tfrac{15}{17}

CAH.

\tan A = \tfrac{8}{15}

TOA.

Your turn, guided

Your turn: legs 7 (opposite) and 24, hypotenuse 25. $\cos A$ :

\tfrac{24}{25}

Adjacent/hypotenuse.

Practice ★

A right triangle has legs 12 and 16 and hypotenuse 20. Angle $A$ is opposite the leg 12. Find $\sin A$ .
A right triangle has legs 6 and 8 and hypotenuse 10. Angle $A$ is opposite the leg 6. Find $\sin A$ .

In a right triangle with acute angle $21^\circ$ , which ratio is $\cos 21^\circ$ ?
1. adjacent ÷ hypotenuse
2. opposite ÷ hypotenuse
3. opposite ÷ adjacent

Watch out!

Practice ★★

Same kind of triangle: legs 15 and 36, hypotenuse 39. Angle $B$ is opposite the leg 36. Find $\tan B$ .
Same kind of triangle: legs 20 and 48, hypotenuse 52. Angle $B$ is opposite the leg 48. Find $\tan B$ .

Exit ticket

Three questions to check you have got it.

A right triangle has legs 12 and 16 and hypotenuse 20. Angle $A$ is opposite the leg 12. Find $\sin A$ .
A right triangle has legs 6 and 8 and hypotenuse 10. Angle $A$ is opposite the leg 6. Find $\sin A$ .
A right triangle has legs 27 and 36 and hypotenuse 45. Angle $A$ is opposite the leg 27. Find $\sin A$ .

Summary card

Trigonometric ratios

SOH-CAH-TOA, for an acute angle of a right triangle.

Watch out: opposite and adjacent depend on which angle you look from.

About this lesson