Evaluate the following:
(i) sin 780 cos 180 – cos 780 sin 180 (ii) cos 470 cos 130 - sin 470 sin 130
(iii) sin 360 cos 90 + cos 360 sin 90 (iv) cos 800 cos 200 + sin 800 sin 200
(i) Given sin 780 cos 180 – cos 780 sin 180
We know that sin(A -B) = sinA cosB - cosA sinB
⇒ sin 780 cos 180 – cos 780 sin 180 = sin(78 – 18) °
= sin 60°
(ii) Given cos 470 cos 130 - sin 470 sin 130
We know that cosA cosB – sinA sinB = cos(A +B)
⇒ cos 470 cos 130 - sin 470 sin 130 = cos(47 + 13) °
= cos 60°
= 1/2
∴ cos 470 cos 130 - sin 470 sin 130 = 1/2
(iii) Given sin 360 cos 90 + cos 360 sin 90
We know that sin(A +B) = sinA cosB + cosA sinB
⇒ sin 360 cos 90 + cos 360 sin 90 = sin(36 + 9) °
= sin 45°
(iv) Given cos 800 cos 200 + sin 800 sin 200
We know that cosA cosB + sinA sinB = cos(A -B)
⇒ cos 800 cos 200 + sin 800 sin 200 = cos(80 - 20) °
= cos 60°
= 1/2
∴ cos 800 cos 200 + sin 800 sin 200 = 1/2