If tan theta 65 pi SolutionTan theta65 and theta is in thir

Solution

Tan theta=6/5    and theta is in third quadrant . And in third quadrant,sine and cosine are negative

here opposite=6   and adjacent=5

hypotenuse=sqrt(36+25)=sqrt61

sin theta= -6/sqrt61      cos theta= -5/sqrt61

a.sin 2 theta=2sin theta cos theta=2(-6/sqrt61)(-5/sqrt61)= 60/61

b. cos 2theta= 2cos²theta-1 = 2(-5/sqrt61)²-1= -11/61

c. sin(theta/2)=sqrt(1-cos theta)/2   = sqrt(1+ 5/sqrt61)/2= sqrt((sqrt61 + 5)/126)

d. cos (theta/2)=-sqrt(1+cos theta)/2= -sqrt(1-5/sqrt61)/2=-sqrt((sqrt61 - 5)/126)