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 .