Trig equations that need an identity first

5 minAS Level1.5.41.5.5

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can spot which identity turns a messy trig equation into one you can solve, solve quadratics in $\sin\theta$ or $\cos\theta$ , and reject the impossible roots.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\alpha$	the principal value: the angle your calculator gives
$\theta$ , $x$	the unknown angle we're solving for

Three moves

You see…	Move	You get
$a\sin x = b\cos x$ with $a\ne0$	Check $\cos x=0$ is not a solution, then divide by $a\cos x$	$\tan x = \tfrac{b}{a}$
$\sin^2$ mixed with $\cos$ (or $\cos^2$ with $\sin$ )	swap with $\sin^2 x = 1 - \cos^2 x$ (or vice versa)	a quadratic in $\sin x$ or $\cos x$
a common factor like $\sin x$ on both sides	bring everything to one side and factorise	two simpler equations

Then it's the solving recipe from before, once per equation.

Worked example: Move 1: sin and cos to tan

Solve $3\sin x = 2\cos x$ for $0^\circ \leqslant x \leqslant 360^\circ$ .

Have a go first?

Quick check

CalculateQuestion 1 · Recall

Answer the question above to continue.