SOLUTION: If cos&#952; = -2/3, and 450 degrees < &#952; < 540 degrees, find: A) Exact value of cos(1/2)&#952; B) Exact value of tan(2&#952;)

Algebra ->  Trigonometry-basics -> SOLUTION: If cos&#952; = -2/3, and 450 degrees < &#952; < 540 degrees, find: A) Exact value of cos(1/2)&#952; B) Exact value of tan(2&#952;)      Log On


   

Question 964583: If cosθ = -2/3, and 450 degrees < θ < 540 degrees, find:
A) Exact value of cos(1/2)θ
B) Exact value of tan(2θ)
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
If cosθ = -2/3, and 450 degrees < θ < 540 degrees, find:
A) Exact value of cos(1/2)θ
B) Exact value of tan(2θ)
***
cos%28x%2F2%29=-sqrt%28%281%2Bcosx%29%2F2%29=-sqrt%28%281-2%2F3%29%2F2%29=-sqrt%281%2F6%29
sinx=sqrt%281-cos%5E2%28x%29%29=sqrt%281-4%2F9%29=sqrt%285%2F9%29=sqrt%285%29%2F3
..
sin(2x)=2sinxcosx=2sqrt(5)/3*-2/3=-(4√5)/9
cos(2x)=cos^2(x)-sin^2(x)=4/9-5/9=-1/9
tan(2x)=sin(2x)/cos(2x)=4√5
..
check:
cosx=-2/3
x=491.81˚
2x=983.62˚
x/2=245.91
cos(x/2)≈cos(245.91)≈-0.4082
exact value as computed above=√(1/6)≈-0.4082
tan(2x)=tan(983.62)≈8.94
exact value as computed above=4√5≈8.94