Sine, Cosine and Tangent

The three primary trigonometric ratios: sine, cosine, and tangent in a right triangle.

Right Triangle Trigonometry

In a right triangle with angle $\theta$ :

Right triangle with angle θ, opposite, adjacent, and hypotenuse sides

SOH-CAH-TOA

\sin\theta = \dfrac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \dfrac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \dfrac{\text{opp}}{\text{adj}}

Key Identities

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

$\sin^2\theta + \cos^2\theta = 1 \quad \text{(Pythagorean identity)}$

Tip

Memory trick: SOH-CAH-TOA — Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Special Angles

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

Find sin, cos, tan for a 3-4-5 right triangle (angle opposite side 4)

Hypotenuse = 5, opposite = 4, adjacent = 3.

$\sin\theta = 4/5 = 0.8$ , $\cos\theta = 3/5 = 0.6$ , $\tan\theta = 4/3 \approx 1.33$

Real-World Applications

Real world: architecture — finding roof height

An architect knows a roof rafter is 8 m long and makes a 30° angle with the horizontal. The vertical rise (height) of the roof is:

$\text{height} = 8 \times \sin 30° = 8 \times 0.5 = 4 \text{ m}$

The horizontal span is: $8 \times \cos 30° \approx 8 \times 0.866 = 6.93 \text{ m}$

Real world: navigation — finding north/east components

A ship sails 50 km on a bearing of 40° from north. The northward component is $50 \times \cos 40° \approx 38.3$ km, and the eastward component is $50 \times \sin 40° \approx 32.1$ km — used to plot the ship's exact position on a chart.

Real world: physics — projectile launch components

A ball is kicked at 20 m/s at an angle of 60° above the ground. The horizontal and vertical velocity components are:

$v_x = 20 \times \cos 60° = 20 \times 0.5 = 10 \text{ m/s}$ (horizontal)

$v_y = 20 \times \sin 60° \approx 20 \times 0.866 = 17.3 \text{ m/s}$ (vertical)

Tip

Where do you see this? Sin and cos split any force, velocity, or distance into horizontal and vertical components — used in engineering, navigation, architecture, and physics every day.

Practice Problems

In a right triangle with hypotenuse 10 and one angle 30°, find the opposite side.

Show answer

opp = 10 × sin(30°) = 10 × 0.5 = 5

Find tan(45°)

Show answer

tan(45°) = 1 (since sin = cos = √2/2)