Question

Find the exact value of  given .

	A.
	B.
	C.
	D.

Find the exact value of  given . 

	A.
	B.
	C.
	D.

 

Answer 1

1) sin(θ) = 5/13

θ is in second quadrant.

sin(θ) = [ Opposite side ]/[ Hypotenuse ]

From the pythageron therom ,

Adjacent side = √[(hypotenuse)² - (opposite side)² ]

= √[(13)² - (5)² ]

= √[169 - 25 ]

= √144

Adjacent side = 12

cos(θ) = [ adjacent side ]/[hypotenuse ]

cos(θ) = - 12/13

Apply the half angle identity cos(θ/2) =  √[(1 + cos(θ))/2]

cos(θ/2) = √[ 1 + (-12/13)]/2

= √[13-12]/2(13)

= √1/2(13)

=  1/√26

=  √26/[√26(√26)]

= √26/26

Option D is correct choice.

Answer 2

2)cos(θ) = -12/13

θ is in third quadrant.

Apply the double angle identity cos(2θ) = 2cos²θ - 1 

= 2 (-12/13)² - 1 

= 2(144/169) - 1

= (288/169) - 1

= ( 288 - 169)/169

= 119/169

Option B is correct choice.