Find sinx2 cosx2 and tanx2 from the given information cos x

Find sin(x/2), cos(x/2), and tan(x/2) from the given information. cos (x) = - 60/61, 180 degree

Solution

cosx= -60/61 x is in QIII

now sin^2 x +cos^2 x =1

sin^2 x = 1- ccos^2 x

sin^2 x = 1 -(60/61)^2

sin^2 x = 1 - 3600 /3721

sin^2 x = 121/3721

sinx = -11/61

cos2x = 1 -2sin^2x

plug x=x/2

then we get cosx = 1 - 2sin^2(x/2)

-60/61 = 1 - 2sin^2 (x/2)

2sin^2(x/2) = 1+60/61

2sin^2(x/2) = 121/61

sin(x/2) = 11/sqrt(122)

now we got sin(x/2) we know sin2x = 2sinxcosx

plug x=x/2

sinx = 2sin(x/2). cos(x/2)

-11/61 = 2 11/sqrt(122) . cos(x/2)

-sqrt(122) /122 = cos(x/2)

cos(x/2) = -1/sqrt(122)

tan(x/2) = sin(x/2) /cos(x/2)

= -11

b). cos(-3pi/8)

Cos 135 = - 1/sqrt(2)

cos(-3pi/8)
= Cos(3 pi/8)
= Cos[(3 pi/4) / 2]
= Cos[135 / 2]
= sqrt[(1 + Cos 135) / 2] >>Using sub-multiple angle relation and taking positive sign only as Cosine value is positive for acute angle
= sqrt[(1 - 1/sqrt{2}) /2]