Answer:
cos(α2)= −1/5√2
Step-by-step explanation:
you are given tan(α)=7/24 , so you can find the length of the hypotenuse using the Pythagorean theorem, which is a2+b2=c2 with a and b being the legs of the triangle and c being the hypotenuse of the triangle.
c2=7^2+24^2
⟹c2=49+576
⟹c2=625
⟹c=25
if π<α<3π/2 , then α has to lie in quadrant III where cosine is negative. so cos(α)= −24/25
The half-angle identity for the cosine function is cos(α2)=±√1+cos(α)/2 , so plugging the information in, we get
cos(α2)=±√ 1+(−24/25)/2
⟹
cos(α2)=±√1/25 / 2
⟹cos(α2)=±√1/50
⟹cos(α2)=±1/5√2
