csc theta=-6/5, tan theta>0

Question

Use the given information to find the exact value of thr trigonometric function cos theta/2

Answer 1

cscθ=-6/5

cscθ=1/sinθ

sinθ=-5/6

tanθ>0 implies quadrant 3

cos=adjacent/hypotenuse

cosθ=-√(11)/6

cos(θ/2)=-√[(1+cosθ)/2]

cos(θ/2)=-√([1-(√11/6)]/2)

cos(θ/2)=-√([(6-√11)/6]/2)

cos(θ/2)=-√([6-√11]/12)

cos(θ/2)=-(1/2)√([6-√11]/3)

Answer 2

If the Csc θ is negative and the Tan θ is positive, then the angle θ must be in quadrant 3. Therefore θ/2 must be in quadrant 2 in which Cosine is negative.

 

Csc θ = -6/5 1/Sin θ

 

sin θ = -5/6

 

θ = 236.44 degrees

 

θ/2 = 118.22

 

Cos θ/2 = -.473