Find the value of the following. (i) 3 sin π/6 sec π/3 - 4 sin 5π / 6 cot π/6

Question

Find the value of the following.

(i) 3 sin\(\frac{\pi}{6}\) sec \(\frac{\pi}{3}\) - 4 sin\(\frac{5 \pi}{6}\) cot\(\frac{\pi}{4}\)

(ii) sin² \(\frac{\pi}{6} + cos^2 \frac{\pi}{3} - tan^2 \frac{\pi}{4}\)

(iii) 2 sin² \(\frac{\pi}{6}\) + cosec²\(\frac{7 \pi}{3}\) cos² \(\frac{\pi}{3}\)

iv) sin 75° 

v) tan 15°

Answer 1

 

iv) sin 75° = sin(45° + 35°) 

= sin 45° cos30° + cos45° sin 30°

v) tan 15° = tan(45° – 30°) = \(\frac{tan 45^\circ-tan 30^\circ}{1+tan 45^\circ tan 30^\circ}\)

\(\cfrac{1-\frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt 3}}\) = \(\frac{\sqrt{3}-1}{\sqrt{3} + 1}\)