Equations like sin 2x = k and cos(x − 30°) = k

5 minAS Level1.5.5

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

By the end of this lesson

By the end you can solve equations such as $5\sin 2x = 2$ , $\cos(x - 30^\circ) = \tfrac12$ or $4\cos 2x + 1 = 0$ in a given interval, without losing (or inventing) solutions.

Change the variable, change the interval

The trick is to solve for the whole angle first. For $\sin 2x = k$ :

Let $u = 2x$ .
Transform the interval: if $0^\circ \leqslant x \leqslant 360^\circ$ , then $0^\circ \leqslant u \leqslant 720^\circ$ .
Solve $\sin u = k$ for every $u$ in the new interval (the recipe from the last lesson).
Convert back: $x = \tfrac{u}{2}$ .

Skip step 2 and you'll lose half the answers. It's the single most common way to drop marks on these.

Quick check

Multiple choiceQuestion 1 · Recall

To solve $\sin 3x = 0.5$ for $0^\circ \leqslant x \leqslant 180^\circ$ , you let $u = 3x$ . What interval must you find $u$ in?

0^\circ \leqslant u \leqslant 540^\circ

0^\circ \leqslant u \leqslant 180^\circ

0^\circ \leqslant u \leqslant 60^\circ

0^\circ \leqslant u \leqslant 360^\circ

Answer the 2 questions above to continue.

Previous lessonSolving sin x = k, cos x = k and tan x = k Next lessonTrig equations that need an identity first

Squash the graph horizontally and twice as many waves fit in the interval, so the line meets it twice as often.

CalculateQuestion 2 · Recall

How many solutions does $\cos 2x = 0.3$ have for $0^\circ \leqslant x \leqslant 360^\circ$ ?