Trigonometry Basics: Sin, Cos, Tan, the Unit Circle, and Key Identities

Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. It underpins everything from navigation and engineering to physics and computer graphics. In this guide, we cover the core trigonometric functions, the unit circle, key identities, and worked examples to build a solid foundation.

The Three Primary Trig Functions

For a right-angled triangle with an angle $\theta$ , the three primary trigonometric functions are defined as ratios of the triangle's sides:

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

A common mnemonic is SOH-CAH-TOA:

Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent

Note that tangent can also be expressed as:

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

The Reciprocal Functions

Each primary function has a reciprocal:

\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}

These appear less frequently in introductory courses but become important in calculus, particularly when integrating trigonometric expressions.

The Unit Circle

The unit circle is a circle of radius 1 centred at the origin. Any point on the unit circle can be written as $(\cos\theta, \sin\theta)$ , where $\theta$ is the angle measured anticlockwise from the positive $x$ -axis.

Key angles and their values:

Angle	Radians	sin	cos	tan
$0°$	$0$	$0$	$1$	$0$
$30°$	$\pi/6$	$1/2$	$\sqrt{3}/2$	$1/\sqrt{3}$
$45°$	$\pi/4$	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
$60°$	$\pi/3$	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$
$90°$	$\pi/2$	$1$	$0$	undefined

Degrees vs Radians

Radians are the standard unit of angle measurement in mathematics. The conversion formulas are:

\text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}

A full revolution is $360° = 2\pi$ radians. A right angle is $90° = \pi/2$ radians.

Fundamental Trigonometric Identities

Pythagorean Identity

The most important identity in trigonometry comes directly from the Pythagorean theorem applied to the unit circle:

\sin^2\theta + \cos^2\theta = 1

Dividing through gives two related identities:

1 + \tan^2\theta = \sec^2\theta, \qquad 1 + \cot^2\theta = \csc^2\theta

Angle Addition Formulas

\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta

\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}

Double Angle Formulas

Setting $\alpha = \beta = \theta$ in the addition formulas:

\sin 2\theta = 2\sin\theta\cos\theta

\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Worked Example 1: Finding a Side Length

Problem: A right-angled triangle has a hypotenuse of length 10 and an angle of $35°$ opposite the unknown side. Find the length of the opposite side.

Using the sine ratio:

\sin 35° = \frac{\text{opposite}}{10}

\text{opposite} = 10 \sin 35° = 10 \times 0.5736 = 5.74

Worked Example 2: Finding an Angle

Problem: A ladder of length 6 m leans against a wall. The foot of the ladder is 2 m from the base of the wall. What angle does the ladder make with the ground?

The adjacent side is 2 and the hypotenuse is 6:

\cos\theta = \frac{2}{6} = \frac{1}{3}

\theta = \cos^{-1}\!\left(\frac{1}{3}\right) \approx 70.53°

Worked Example 3: Using an Identity

Problem: If $\sin\theta = 3/5$ and $\theta$ is in the first quadrant, find $\cos\theta$ and $\tan\theta$ .

From the Pythagorean identity:

\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{9}{25} = \frac{16}{25}

\cos\theta = \frac{4}{5} \quad \text{(positive in Q1)}

\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4}

The Law of Sines and Law of Cosines

For any triangle (not just right-angled) with sides $a, b, c$ opposite angles $A, B, C$ :

Law of Sines

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

The law of cosines generalises the Pythagorean theorem. When $C = 90°$ , the formula reduces to $c^2 = a^2 + b^2$ .

Applications of Trigonometry

Navigation: Bearing calculations and GPS positioning rely on trigonometric functions.
Physics: Resolving forces into components, analysing waves and oscillations, and computing projectile trajectories.
Engineering: Structural analysis, signal processing (Fourier transforms), and AC circuit analysis.
Computer graphics: Rotating objects, calculating lighting angles, and mapping textures onto 3D surfaces.

Try It Yourself

Explore trigonometric calculations with our free Triangle Calculator for solving any triangle, or use our Scientific Calculator to evaluate trig functions for any angle.