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“Trigonometric Functions: Exam Practice Pack” is available to Standard subscribers.

What the exams add to this chapter: solving trigonometric equations, the greatest and least values of an expression, and identities in a triangle, each with worked examples, and a full timed paper.

3Lessons11Problems2Python Labs1Question Set1Mock Test

Read the first lesson freeSolving trigonometric equations

What's inside

Solving trigonometric equations: An equation such as $\sin x = \frac{1}{2}$ has infinitely many solutions, because sine repeats. The principal solutions lie in one turn, from $0$ up to $2\pi$

The chapter itself stays free

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Greatest and least values: The greatest value of

a\sin x + b\cos x

is not

a + b

, because the two parts peak at different angles. Written as one wave,

R\sin(x + c)

with

R = \sqrt{a^2 + b^2}

, its values run from

-R

R

. Products become single waves through the double angle formulas. An expression in

\sin x

alone becomes a quadratic in

t = \sin x

with

t

from

-1

1

Identities in a triangle: In a triangle

A + B + C = \pi

, so each angle is

\pi

minus the other two. That gives

\sin C = \sin(A + B)

and

\tan C = -\tan(A + B)

, and from them identities that hold in every triangle, such as

\tan A + \tan B + \tan C = \tan A\tan B\tan C

and

\sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C

. They find a missing angle in one line.