o cos x + cos y = 2 cos( x + y)/ 2 cos( x – y)/ 2 o cos x – cos y = – 2sin( x + y)/ 2 sin( x- y)/ 2 o sin x + sin y = 2sin( x + y)/ 2 cos( x- y)/ 2 o sin x – sin y = 2cos( x + y)/ 2 sin( x- y)/ 2
Numerical: Find the value of sin 15 °. Solution:
sin 15 ° = sin( 45 ° – 30 °) = sin 45 ° cos 30 ° – cos 45 ° sin 30 ° = 1 / √2 * √3 / 2- 1 / √2 * ½ =( √3-1)/ 2√2
Numerical: Prove that sin( x + y) / sin( x-y) =( tan x + tan y)/ tan x- tan y)
Solution:
sin( x + y) / sin( x-y) =( sin x * cos y + cos x * sin y) /( sin x * cos y- cos x * sin y)
Divide numerator & denominator by cos x cos y, to get sin( x + y) / sin( x-y) =( tan x + tan y)/ tan x- tan y)
Principal and General Solution Principal solutions: The solutions where 0 ≤ x < 2π.
General solution: The expression involving integer‘ n’ this gives all solutions of a trigonometric equation. To derive general solution we will use the fact that:
o o
Values of sinx repeat after an interval of 2π Values of cos x repeat after an interval of 2π o Values of tanx repeat after an interval of π.