Question

If sinx+siny = a and cosx +cosy = b find the value of sin x-y/2 in terms of a and b

Answer 1

sinx + siny = a

squaring both sides

(sinx + siny)^2 = a^2 --------------(1)

cosx + cosy = b

squaring both sides

(cosx + cosy)^2 = b^2 ----------(2)

add eq (1) and (2)

(sinx + siny)^2 + (cosx + cosy)^2 = b^2 + a^2

sin^2x + sin^2y + 2sinxsiny + cos^2x + cos^2y + 2cosxcosy = a^2 + b^2

sin^2x + cos^2x + sin^2y + cos^2y + 2sinxsiny + 2cosxcosy = a^2 + b^2

1 + 1 + 2(sinxsiny + cosxcosy) = a^2 + b^2

2 + 2cos(x-y) = a^2 + b^2

We can write 2 as 4 - 2

4 - 2 + 2cos(x-y) = a^2 + b^2

4 - 2{1 - cos(x-y)} = a^2 + b^2

we know that

2sin^2A = 1 - cos2A

2sin^2A/2 = 1 - cosA

so,

4 - 2{2sin^2 x-y/2 } = a^2 + b^2

2{2sin^2 x-y/2 } = 4 - a^2 - b^2

4sin^2 x-y/2 = 4 - a^2 - b^2

sin^2 x-y/2 = (4 - a^2 - b^2) /4

sin x-y/2 = 1/2✓(4 - a^2 - b^2)

Answer 2

Answer:

hope it helps you and don't forget to follow mè