(a) Prove that for any real numbers and , the equality holds if and only if for some…

MATHEMATICS · Subjective · A Level · Mathematics · Trigonometry · Trigonometry

Problem

(a) Prove that for any real numbers

\alpha

and

\beta

, the equality

\sin\alpha = \sin\beta

holds if and only if

\alpha = n\pi + (-1)^{n}\beta

for some integer

n

. (b) Using the result of part (a), obtain the general solution of the equation

\sin(3x) = \sin\left(7x + \dfrac{\pi}{4}\right)

. (c) List all principal values of

x

in degrees that satisfy the above equation in the interval

0^{\circ} \le x < 360^{\circ}