Question

If sin 3x = 1 and 0° S 3x < 90°, find the values of
(i) sin x
(ii) cos 2x
(iii) tan^2x-sec^2x.
​

Answer 1

Answer:

Given :

Sin (3x) = 1 where. 0°<3x <90°

To find :

(i) sin x

(ii) cos 2x

(iii) tan^2x-sec^2x.

Solution :-

We know that if

$\sin(\theta) = 1 \\ \theta \: = 90 \: \: \: \: \: \:$

then replace. theta by 3x

$\sin(3x) = 1 \\ \\ 3x = 90 \: \: \: \: \: \: \: \\ \\ x = \frac{90}{3} = 30$

Hence. x=30°

$(1) \sin(x) = \sin(30) \\ \\ \: \: \: \: \: \: \: \: \: \: = \frac{1}{2}$

$(2) \: \cos(2x) = \cos(2 \times 30) \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \cos(60) \\ \\ = \frac{1}{2}$

${ \tan^{2} (2x) } - \sec {}^{2} (2x) \\ \\ = \tan {}^{2} (2 \times 30) - \sec {}^{2} (2 \times 30) \\ \\ = \tan {}^{2} (60) - \sec {}^{2} (60) \\ \\ = ( { \sqrt{3} )}^{2} - {2}^{2} \\ \\ = 3 - 4 \\ \\ = - 1$