Trigonometric Equations

Revise Trigonometric Equations for Pure Mathematics 2 WMA12 (AS Level) — revision notes and instant AI marking.

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In a Nutshell

Trigonometric identities allow you to rewrite equations in forms that are easier to solve (e.g., converting everything to one trig function).
The CAST diagram shows which trig functions are positive in each quadrant, helping you find all solutions in a given range.
Linear equations (like sin x = 0.5) are solved using the CAST diagram or graph sketching.
Quadratic equations (like 2sin²x + 3sin x − 2 = 0) are solved by substitution and factorising, but you must check that solutions actually exist.
Transformed equations (like sin(2x) or cos(x − 30°)) require you to change the range first, solve, then transform solutions back.
Your calculator only gives the principal value — you must find all other solutions yourself using periodicity and the CAST diagram.

Simple Trigonometric Identities

What is a Trigonometric Identity?

A trigonometric identity is a mathematical statement that is true for all values of x or θ. Unlike an equation (which is true only for certain values), an identity is always true. These identities allow you to simplify complex trig equations before solving them.

The Two Essential Identities You Must Know

Identity 1: Tangent Definition

tan θ = sin θ / cos θ

Tangent is defined as sine divided by cosine. This is not derived from a right-angled triangle, but from the unit circle: tan is the gradient of the radius at angle θ.

Identity 2: Pythagoras for Trig

sin²θ + cos²θ = 1

This comes directly from Pythagoras' theorem applied to the unit circle (where the hypotenuse is 1). You can rearrange this as sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ.

Where Do These Identities Come From?

For the tangent identity: From the definitions sin θ = OPP/HYP, cos θ = ADJ/HYP, and tan θ = OPP/ADJ, if we divide sin by cos we get (OPP/HYP) ÷ (ADJ/HYP) = OPP/ADJ = tan θ. ✓

For Pythagoras identity: The unit circle has equation x² + y² = 1. Since x = cos θ and y = sin θ on the unit circle, we get sin²θ + cos²θ = 1. ✓

How to Use Trigonometric Identities

Identities are used to:

Simplify equations — convert everything to one trig function so it becomes solvable.
Prove statements — show one side of an equation is identical to the other.
Rearrange and manipulate — build equivalent forms that make the equation easier to work with.

Exam tip: These identities are not in the formula book — you must memorise them. If you see tan disappear from an equation, someone's used tan = sin/cos. If sin² or cos² appears in a single equation, try substituting 1 − cos² or 1 − sin².

Example 1: For an angle x, 5cos x = 2sin x. Find tan x.

Rearrange to isolate tan x:
Divide both sides by cos x:
5cos x / cos x = 2sin x / cos x
5 = 2 tan x

Solve:
tan x = 5/2 = 2.5

Try it yourself

If sin x = 3cos x, find the exact value of tan x.

Linear Trigonometric Equations

What is a Linear Trigonometric Equation?

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Also in the full note

Quadratic Trigonometric Equations
Strategy for Trigonometric Equations
What to Memorise
Concepts Checklist
Exam Tips & Common Pitfalls
The CAST Diagram: Your Roadmap to All Solutions
Solving Equations with Transformed Arguments
What is a Quadratic Trigonometric Equation?

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Trigonometric Equations