Right Triangle w/ Angle Question

Given: A right triangle with sides of lengths 8, 15, and 17. Find the exact value of cos θ. Find the exact value of sin (θ/2).

And in order to understand this problem, could you explain step-by-step as to how you solved it successfully? Thanks.

Which angle of the triangle is θ?

Let's assume that θ is the angle opposite the side of length 15. The cos θ = 8/17.

sin (θ/2) = √((1 - 8/17)/2) = √(9/34) = 3/√34 = 3√34/34

It depends what angle you are talking about. Depending on the angle, cosθ = 8/17 or cosθ = 15/17 (since cosθ = adj/hyp, where adj is the side adjacent to θ and hyp is the hypotenuse).

At this point, you can apply the half-angle formula for sine:
sin(θ/2) = ±√[(1 - cos θ)/2].

Since θ is an angle of the triangle, you know that:
0° < θ < 90° ==> 0° < θ/2 < 45°,

so θ/2 is in Quadrant I and sin(θ/2) > 0. Thus, pick the positive sign to yield:
sin(θ/2) = √[(1 - cos θ)/2].

Plug in your value of cosθ and plug-and-chug!

I hope this helps!

Assume θ is the smallest angle.
cos θ = 15/17
sin (θ/2) = sqrt[(1-cos θ)/2] = sqrt[(1 - 15/17)/2] = sqrt(17)/17