Topic 4 of 7

Right triangles and basic trigonometry

Step 12 · Geometry topics

Use the side ratios of 45°-45°-90° and 30°-60°-90° triangles, define sine, cosine and tangent, and use them to find missing sides and angles of right triangles.

Special right triangles

Two right triangles turn up so often that their side ratios are worth knowing by heart.

Triangle	Sides	Example
45°-45°-90°	x, x and x√2	legs 5 and 5, hypotenuse 5√2
30°-60°-90°	x, x√3 and 2x	short leg 4, long leg 4√3, hypotenuse 8

Example: A 45°-45°-90° triangle

A square has sides of 6. How long is its diagonal?

The diagonal cuts it into two 45°-45°-90° triangles with legs of 6.
The hypotenuse is 6√2, about 8.49.

Answer6√2

Example: A 30°-60°-90° triangle

The hypotenuse of a 30°-60°-90° triangle is 10. How long are the legs?

The short leg is half the hypotenuse: 5.
The long leg is 5√3, about 8.66.

Answer5 and 5√3

Sine, cosine and tangent

In a right triangle, the ratios of the sides depend only on the angle. For an angle θ, name the side opposite θ, the side adjacent to it (next to it, but not the hypotenuse) and the hypotenuse.

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}

A right triangle with legs 8 and 6 and hypotenuse 10. For θ at the left corner: opposite 6, adjacent 8, hypotenuse 10.

Example: Finding the ratios

In that triangle, find sin θ, cos θ and tan θ.

sin θ = $\frac{6}{10}$ , which is 0.6.
cos θ = $\frac{8}{10}$ , which is 0.8.
tan θ = $\frac{6}{8}$ , which is 0.75.

Answer0.6, 0.8 and 0.75

Solving right triangles

To find a missing side, choose the ratio that links the side you know with the side you want. To find a missing angle, use the inverse: sin⁻¹, cos⁻¹ or tan⁻¹.

Example: A missing side

A ramp makes a 12° angle with the ground and is 5 m long. How high does it rise? (sin 12° ≈ 0.208)

The rise is opposite the angle and the ramp is the hypotenuse: sin 12° = rise ÷ 5.
rise = 5 × 0.208 = 1.04.

AnswerAbout 1.04 m

Example: A missing angle

A right triangle has the side opposite θ equal to 7 and the adjacent side equal to 7. What is θ?

tan θ = 7 ÷ 7 = 1.
tan⁻¹(1) = 45°.

Answer45°

Example: Angle of elevation

From 30 m away, the top of a tree is at an angle of elevation of 40°. How tall is the tree? (tan 40° ≈ 0.839)

tan 40° = height ÷ 30.
height = 30 × 0.839, about 25.2.

AnswerAbout 25.2 m

Try it yourself

Work each one on paper first, then open the answer to check.

A 45°-45°-90° triangle has legs of 8. How long is the hypotenuse?
Show answer
8√2, about 11.3
A 30°-60°-90° triangle has a short leg of 7. How long is the hypotenuse?
Show answer
14
For angle θ, the opposite side is 5, the adjacent side 12 and the hypotenuse 13. What is cos θ?
Show answer
$\frac{12}{13}$
What is tan θ in the same triangle?
Show answer
$\frac{5}{12}$
A 10 ft ladder makes a 70° angle with the ground. How high does it reach? (sin 70° ≈ 0.940)
Show answer
About 9.4 ft
For angle θ, the opposite side is 3 and the hypotenuse is 6. What is θ?
Show answer
30°, because sin θ = 0.5.