Trigonometry Basics – Understanding Triangles

Learn how to use sine, cosine, and tangent to find unknown sides and angles in right triangles.

Why Trigonometry?

Trigonometry lets us find unknown sides or angles in right triangles without measuring directly.

The Three Ratios

In a right triangle with angle $\theta$ :

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Tip

Memory: SOH-CAH-TOA

Finding a Missing Side

If you know one angle (other than the right angle) and one side, you can find the others.

1
Label the sides
Identify opposite, adjacent, and hypotenuse relative to your angle.
2
Choose the correct ratio
Which two sides are involved? Pick sin, cos, or tan accordingly.
3
Set up the equation and solve
Rearrange to find the unknown side.

Right triangle: angle 40°, hypotenuse 10 cm. Find the opposite side.

$\sin 40° = \text{opp}/10$

$\text{opp} = 10 \sin 40° \approx 10 \times 0.643 = 6.43 \text{ cm}$

Finding a Missing Angle

Use inverse trig functions: $\theta = \arcsin(x)$ , $\arccos(x)$ , or $\arctan(x)$ .

Opposite = 5, adjacent = 12. Find the angle.

$\tan\theta = 5/12$

$\theta = \arctan(5/12) \approx 22.6°$

Real world: measuring a building's height

A surveyor stands 40 m from the base of a tower. The angle of elevation to the top is 28°. Using tangent: $\text{height} = 40 \times \tan 28° \approx 40 \times 0.532 = 21.3\text{ m}$ . Trigonometry lets engineers measure heights that are impractical to climb.

Real world: wheelchair ramp design

A ramp must rise 0.75 m over a horizontal distance of 6 m. The angle of incline: $\theta = \arctan\!\left(\frac{0.75}{6}\right) \approx 7.1°$ . Building regulations specify a maximum angle (typically ≤ 5°), so trigonometry ensures accessible design.

Real world: navigation and GPS

A ship travels 50 km on a bearing of N 30° E. The northward component $= 50 \cos 30° \approx 43.3\text{ km}$ and the eastward component $= 50 \sin 30° = 25\text{ km}$ . GPS systems use these trigonometric decompositions to compute exact position changes.

Tip Remember SOH-CAH-TOA. Label opposite, adjacent, and hypotenuse relative to the angle you are working with before choosing a ratio.

Practice Problems

Triangle: hypotenuse = 15, angle = 30°. Find adjacent side.

Show answer

adj = 15 cos(30°) = 15 × (√3/2) ≈ 13.0

Opposite = 3, hypotenuse = 5. Find the angle.

Show answer

θ = arcsin(3/5) = arcsin(0.6) ≈ 36.9°

Find tan(60°)

Show answer

tan(60°) = √3 ≈ 1.732