Answer the following questions regarding the periodicity and sign of the secant function.

MATHEMATICS · Multi-part · CBSE · Mathematics · Trigonometric Functions · Trigonometric Functions

Problem

Answer the following questions regarding the periodicity and sign of the secant function. (i) Prove that

\cos(\theta+2\pi k)

and hence

\sec(\theta+2\pi k)

have the same sign as

\cos\theta

and

\sec\theta

respectively, for every integer

k

. (ii) Using part (i), find all integers

k

for which

\sec\!\left(\dfrac{\pi}{3}+2\pi k\right)

is negative. (iii) Within one complete revolution

0<\theta<2\pi

, determine all intervals of

\theta

for which both

\sin\theta

and

\sec\theta

are negative simultaneously. Justify your answer with quadrant analysis.