How do you find all solutions of \cos x=k?

x=\pm\alpha, then add multiples of 360^\circ (2\pi).

How do you find all solutions of \tan x=k?

x=\alpha, then add multiples of 180^\circ (\pi).

Solve \sin\theta=0.6 for 0^\circ\le\theta\le360^\circ.

\theta=36.9^\circ and 143.1^\circ.

Solve \cos x=-\frac12 for 0\le x<2\pi.

x=\frac{2\pi}3 and \frac{4\pi}3.

For \cos(x+30^\circ)=\frac12 with -180^\circ<x<180^\circ, what is the interval for x+30^\circ?

-150^\circ<x+30^\circ<210^\circ.

What do you do to the interval when solving \tan2x=1?

Double it, solve for 2x, then halve the answers.

What does \sin^2y=\frac12 imply?

\sin y=\pm\frac1{\sqrt2}, so include both signs.

How do you solve 6\cos^2x+\sin x-5=0?

Use \cos^2x=1-\sin^2x, factorise (3\sin x+1)(2\sin x-1)=0.

Why should you not divide an equation by \cos x?

It loses the solutions where \cos x=0; factorise instead.

Solving trigonometric equationsEdexcel International A Level Maths: Flashcards

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How do you find all solutions of $\sin x=k$ from the principal value $\alpha$?

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How do you find all solutions of $\sin x=k$ from the principal value $\alpha$ ?: $x=\alpha$ or $180^\circ-\alpha$ ( $\pi-\alpha$ ), then add multiples of $360^\circ$ ( $2\pi$ ).
How do you find all solutions of $\cos x=k$ ?: $x=\pm\alpha$ , then add multiples of $360^\circ$ ( $2\pi$ ).
How do you find all solutions of $\tan x=k$ ?: $x=\alpha$ , then add multiples of $180^\circ$ ( $\pi$ ).
Solve $\sin\theta=0.6$ for $0^\circ\le\theta\le360^\circ$ .: $\theta=36.9^\circ$ and $143.1^\circ$ .
Solve $\cos x=-\frac12$ for $0\le x<2\pi$ .: $x=\frac{2\pi}3$ and $\frac{4\pi}3$ .
For $\cos(x+30^\circ)=\frac12$ with $-180^\circ<x<180^\circ$ , what is the interval for $x+30^\circ$ ?: $-150^\circ<x+30^\circ<210^\circ$ .
Solve $\cos(x+30^\circ)=\frac12$ for $-180^\circ<x<180^\circ$ .: $x=30^\circ$ and $-90^\circ$ .
Solve $\tan2x=1$ for $90^\circ<x<270^\circ$ .: $x=112.5^\circ$ and $202.5^\circ$ .
What do you do to the interval when solving $\tan2x=1$ ?: Double it, solve for $2x$ , then halve the answers.
Solve $\sin\left(x+\frac\pi2\right)=\frac34$ for $0<x<2\pi$ to 3 s.f.: $x=0.723$ and $5.56$ .
What does $\sin^2y=\frac12$ imply?: $\sin y=\pm\frac1{\sqrt2}$ , so include both signs.
How do you solve $6\cos^2x+\sin x-5=0$ ?: Use $\cos^2x=1-\sin^2x$ , factorise $(3\sin x+1)(2\sin x-1)=0$ .
Why should you not divide an equation by $\cos x$ ?: It loses the solutions where $\cos x=0$ ; factorise instead.

Exam questions on Solving trigonometric equations

$\sin\theta=0.6$ , for $0^\circ\le\theta\le360^\circ$ .
Hence solve $\sin(\theta-20^\circ)=0.6$ for $0^\circ\le\theta\le360^\circ$ .2 marks
In this question, $0\le x<2\pi$ and angles are in radians.
Solve $\sin\left(x+\frac\pi2\right)=\frac34$ , giving your answers to 3 significant figures.2 marks
Give angles in degrees, with non-exact answers to 1 decimal place.
Solve $\cos(x+30^\circ)=\frac12$ for $-180^\circ<x<180^\circ$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).