If tan(theta) = 8/5, pi <theta< 3pi/2

Question

find the exact value of each of the following:?

 

(a) sin(2theta) (b) cos(2theta) (c) sin(theta/2) (d) cos(theta/2)

 

Answer 1

tan(θ) = 8/5

tan(θ) = opposite side/adjacent side

From the Pythagorean theorem

Hypotenuse = √[(opposite side)² + (adjacent side)²]

= √[ 64 + 25] = √89

θ is in third quadrant.

sin(θ) = opposite side/Hypotenuse

= -8/√89

cos(θ) =  adjacent side/Hypotenuse

= - 5/√89

a) Sin(2θ) = 2sin(θ)cos(θ)

= 2[-8/√89] [-5/√89]

Sin(2θ) = 80/89.

 

b) cos(2θ) = cos²(θ) - sin²(θ)

cos(2θ) = [- 5/√89]² - [8/√89 ]²

= (25/89) - (64/89)

cos(2θ) = - 39/89.

 

c) sin(θ/2) = ± √[1 - cosθ)/2]

= ± √[1 -(- 5/√89)/2]

= ± √{[(1 + (5/√89)]/2}

sin(θ/2) = ± √(√89 + 5/2√89).

 

d) cos(θ/2) = ± √[1 + cosθ)/2]

= ± √[1 +(- 5/√89)/2]

= ± √{[(1 - (5/√89)]/2}

cos(θ/2)  = ± √(√89 - 5)/2√89).