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What is High School / Foundations

Trigonometry?

The mathematics of angles, triangles, and periodic waves — the language of circles, oscillations, and everything that repeats.

Simple Definition

Trigonometry studies the relationships between the angles and sides of triangles. Given one angle and one side of a right triangle, you can find everything else. That's enormously useful in navigation, architecture, physics, and engineering.

But trig goes far beyond triangles. The six trigonometric functions — sine, cosine, tangent, and their reciprocals — are periodic functions that describe waves, rotations, and oscillations. Sound, light, electricity, tides — all modelled with trig.

At its core, trigonometry is about connecting angles to ratios. The angle uniquely determines the ratio of sides in any right triangle of that shape, regardless of size.

The Three Core Functions

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SOH-CAH-TOA

In a right triangle with angle \(\theta\): Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. This mnemonic is the entry point to all of trigonometry.

sin θ

\(\dfrac{\text{opposite}}{\text{hypotenuse}}\)

SOH

cos θ

\(\dfrac{\text{adjacent}}{\text{hypotenuse}}\)

CAH

tan θ

\(\dfrac{\text{opposite}}{\text{adjacent}}\)

TOA

Intuition

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Sine is a wave

Imagine a point travelling around the unit circle at constant speed. Its height above the x-axis traces a perfect sine wave. Its horizontal position traces a cosine wave. Trig functions are the mathematics of circular motion — and since circular motion creates waves, trig is the mathematics of all waves.

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Trig is everywhere

GPS uses trig to triangulate positions. Architects use it to calculate roof angles and structural loads. Physicists use it to decompose forces into components. Electrical engineers use it to analyse AC circuits. Musicians (unconsciously) hear trig — every musical tone is a sine wave.

Visual — Interactive Unit Circle

Move your mouse over the canvas to change the angle — sin and cos update live

Key Identities & Values

Pythagorean	\(\sin^2\theta + \cos^2\theta = 1\)
Tangent	\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
Reciprocals	\(\csc\theta = \frac{1}{\sin\theta},\quad \sec\theta = \frac{1}{\cos\theta},\quad \cot\theta = \frac{1}{\tan\theta}\)
sin at 0°,30°,45°,60°,90°	\(0,\ \frac{1}{2},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{3}}{2},\ 1\)
Derivatives	\((\sin x)' = \cos x,\quad (\cos x)' = -\sin x\)
Double angle	\(\sin 2\theta = 2\sin\theta\cos\theta\)

Quick Examples

Example 1 — Right triangle

Hyp = 10, angle = 30°

Opposite \(= 10\sin 30° = 10 \cdot \frac{1}{2} = 5\)
Adjacent \(= 10\cos 30° = 5\sqrt{3}\)

Example 2 — Identity

Simplify \(\dfrac{\sin^2 x}{\cos^2 x} + 1\)

\(= \tan^2 x + 1 = \sec^2 x\)
(Pythagorean identity variant)

Example 3 — Derivative

\(\dfrac{d}{dx}[\sin(3x)]\)

Chain rule:
\(= 3\cos(3x)\)

Example 4 — Integral

\(\displaystyle\int \cos x\,dx\)

\(= \sin x + C\)

Common Mistakes

✗ sin(a + b) = sin(a) + sin(b)

Sine does not distribute over addition. The correct formula is \(\sin(a+b) = \sin a \cos b + \cos a \sin b\). This is the angle addition formula — memorise it, or at least know it exists.

✗ Confusing degrees and radians

Calculus always uses radians. \(\frac{d}{dx}[\sin x] = \cos x\) is only true when \(x\) is in radians. In degrees the formula picks up a factor of \(\pi/180\). Set your calculator to radians before doing any calculus with trig.

✗ sin⁻¹(x) = 1/sin(x)

\(\sin^{-1}(x)\) means the inverse sine (arcsin), not the reciprocal. \(1/\sin(x)\) is cosecant (\(\csc x\)). The notation is genuinely ambiguous — always check context.

Related Concepts

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Trig is tested across High School and Calculus quizzes on BUders.

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