I am trying to use the half-angle formulas, but I'm just stuck.

Question

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.
7π/12

5 sin(7π/12) =


9 cos(7π/12) =

5 tan(7π/12) =

Answer 1

 

7∏/12=7*180/12=7*15=105º

5sin(105º)=5sin(210/2)

sin(210/2)=±√(1-cos210)/2

±√(1-[-√3/2])/2

±√(1+√3/2)/2

±√[(2+√3)/2]/2

±√[(2+√3)/4]

±√(2+√3)/2 (the second 2 is not inside the √ symbol)

now multiply by 5

(5/2)√(2+√3)

 

9cos(105º)=9cos(210/2)

cos(210/2)=±√(1+cos210)/2

±√(1-√3/2)/2

±√[(2-√3)/2]/2

±√[2-√3]/4

±√(2-√3)/2 (the second 2 is not inside the √ symbol)

now multiply by 9

-(9/2)√(2-√3)

 

5tan(210/2)=±5√[(2+√3)/2]/[(2-√3)/2]=

±5√[(2+√3)/(2-√3)]

rationalize the denominator

±5√(1/[7-4√3]

-5/√(7-4√3)