Introduction to Trigonometric Equations

A trigonometric equation is an equation that contains one or more trigonometric functions of an unknown angle. Unlike trigonometric identities (which are true for all values), trigonometric equations are only satisfied by specific angles within a given domain.

Trigonometric Equation

An equation involving trigonometric functions (sin, cos, tan, etc.) that is true only for particular values of the variable, usually restricted to a given interval.

The general strategy for solving any trig equation follows these steps:

Isolate the trigonometric function
Find the reference angle using the inverse function
Identify all solutions in the given interval using knowledge of the CAST diagram
Check that all solutions lie within the stated domain

Exam Tip

Always read the domain carefully before you start. Solutions outside the given interval score no marks, and missing solutions within it will cost you marks.

Introduction to Trigonometric Equations

Exact Values — Essential for Paper 1

IB SL Paper 1 is a non-calculator paper. When solving trigonometric equations on Paper 1, you must recognise exact values immediately. The table below lists all required exact values.

Angle	$sin$	$cos$	$tan$
$0$	$0$	$1$	$0$
$\frac{π}{6}$ (30°)	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{3}$
$\frac{π}{4}$ (45°)	$\frac{2}{2}$	$\frac{2}{2}$	$1$
$\frac{π}{3}$ (60°)	$\frac{3}{2}$	$\frac{1}{2}$	$3$
$\frac{π}{2}$ (90°)	$1$	$0$	undefined

Exam Tip

When you isolate a trig function and get a value from the table above, the reference angle is exact. For example, $cos x = \frac{1}{2}$ immediately gives reference angle $\frac{π}{3}$ . Always give exact answers on Paper 1 unless told otherwise.

Note

$\frac{2}{2}$ is sometimes written as $\frac{1}{2}$ — these are equivalent. Similarly, $\frac{1}{3} = \frac{3}{3}$ .

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SL 3.8—Solving trig equations

Introduction to Trigonometric Equations

Exact Values — Essential for Paper 1

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