Find the values of the following: (i) sin 76° cos 16° – cos 76° sin 16° (ii) sin π/4 cos π/12 + cos π/4 sin π/12

Question

Find the values of the following:

(i) sin 76° cos 16° – cos 76° sin 16° 

(ii) sin\(\frac{\pi}{4}\)cos\(\frac{\pi}{12}\) + cos\(\frac{\pi}{4}\)sin\(\frac{\pi}{12}\)

(iii) cos 70° cos 10° – sin 70° sin 10° 

(iv) cos² 15° – sin² 15

Answer 1

(i) Given that, sin 76° cos 16° – cos 76° sin 16° 

(∴ This is of the form sin(A – B)) 

= sin(76° – 16°) 

= sin 60°

= \(\frac{\sqrt{3}}{2}\)

(ii) This is of the form sin(A + B) = sin(\(\frac{\pi}{4}+\frac{\pi}{12}\))

= sin(\(\frac{3\pi+\pi}{12}\))

= sin\(\frac{4\pi}{12}\)

= sin\(\frac{\pi}{3}\)

= \(\frac{\sqrt{3}}{2}\) (∵ sin 60° = \(\frac{\sqrt{3}}{2}\))

(iii) Given that cos 70° cos 10° – sin 70° sin 10° 

(This is of the form of cos (A + B), A = 70°, B = 10°)

= cos (70° + 10°) 

= cos 80°

(iv) cos² 15° – sin² 15° 

[∵ cos 2A = cos² A – sin² A, Here A = 15°] 

= cos (2 x 15°) 

= cos 30°

= \(\frac{\sqrt{3}}{2}\)