given that sin a=-8/17, a is in QIV, find sin (a/2), and cos(a/2)?

Question

a=angle
no calculators please

Answer 1

the triangle with sin(a) is -8,15,17

Half angel formula
sin^2(a/2)= 1/2(1-cos(a))= 1/2(1-15/17)=1/17
sin(a/2)=sqrt(1/17)
cos^2(a/2)=1/2(1+cos(a))=1/2(1+15/17)=8/17
cos(a/2)=sqrt(8/17)

Answer 2

Given that a is in Q IV, A/2 must be in Q II.
17^2 - 8^2 = 289 - 64 = 225 = 15^2
cos(a) = 15/17
cos^2(a/2) = (1/2)(1 + cos(a))
cos^2(a/2) = (1/2)(1 + 15/17)
cos^2(a/2) = (1/2)(17 + 15)/17
cos^2(a/2) = (1/2)(32/17) = 16/17
cos(a/2) = - 4√(1/17)

sin^2(a/2) = (1/2)(1 - 15/17)
sin^2(a/2) = (1/2)(17 - 15)/17
sin(a/2) = + √(1/17)

Answer 3

Since a is in IV quadrant, only cos(a) and sec (a) are positive.

Since a is between 3pi/2 and 2pi, a/2 will be in between

3pi/4 and pi, that is 2nd quadrant, where only sin(a/2) and csc(a/2) will be positive.

cos(a) = sqrt(cos^2(a)

cos(a) = sqrt(1-sin^2(a)) = sqrt(1 - (8/17)^2)

=> sqrt(225/289)= 15/17

cos(a) = 15/17

cos(a) = 2cos^2(a/2) - 1 or 1 - 2sin^2(a/2)

calculate cos(a/2)

2cos^2(a/2) - 1 = 15/17

2cos^2(a/2) = 15/17 + 1 = 32/17

cos^2(a/2) = 16/34

cos(a/2) = - 4/sqrt(17) (since cos(a/2) is negative)

calculate sin(a/2)

1 - 2sin^2(a/2) = 15/17

2sin^2(a/2) = 1 - (15/17) = 2/17

sin^2(a/2) = 1/17

sin(a/2) = 1/sqrt(17) (since sin(a/2) is positive)

so
sin(a/2) = 1/sqrt(17) and

cos(a/2) = - 4/sqrt(17)

Answer 4

given that sin a=-8/17, a is in QIV, find sin (a/2), and cos(a/2)?

sin(2(a/2)) = 2 sin(a/2)cos(a/2) = -8/17

and

sin²(a/2) +cos²(a/2) =1

2 equations to solve to find sin(a/2) and cos(a/2)

Answer 5

Drawing the triangle in the fourth quadrant and computing the missing side shows that the side adjacent the angle is -15.

So, cos(a) = -15/17.

Now, using the half-angle formulas:

sin(a/2) = -Sqrt((1-cos(a))/2)
cos(a/2) = -Sqrt(1+cos(a))/2)


sin(a/2) = -Sqrt((1-(-15/17))/2)
cos(a/2) = -Sqrt(1+(-15/17))/2)

sin(a/2) = -Sqrt((32/17)/2)
cos(a/2) = -Sqrt((2/17)/2)

sin(a/2) = -Sqrt(16/17)
cos(a/2) = -Sqrt(1/17)

sin(a/2) = -4/Sqrt(17)
cos(a/2) = -1/Sqrt(17)