Evaluate the following composite expression without using a calculator.

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Evaluate the following composite expression without using a calculator. (a) Simplify the expression

S = \sin\left(-\tfrac{7\pi}{6}\right) + 2\cos\left(-\tfrac{13\pi}{4}\right) - 3\tan\left(-\tfrac{5\pi}{3}\right)

so that no trigonometric function contains a negative angle and all angles lie in

[0,2\pi)

(for

\tan

, use its least positive period). (b) Hence, find the exact numerical value of

S

. (c) Show that each individual trigonometric number obtained in part (b) lies within the correct theoretical range of its function.