I'm having trouble with several questions listed below:

Question

1. Find the exact value of cos(t/2) if sin(t) = -7/25 and pi < t < 3pi/2 

 

2. Find all the solutions in the interval [0,360] 

      cos(2x)= - sqrt3/2 

 

Answer 1

1)

 

Use the half-angle identity for cosine.

 

cos((1/2)t) = √((1 + cost) / 2)

 

 

sine is negative in the 3rd quadrant. Using Pythagorean theorem to find the value of cost,

 

cost = -√[1 - (-7/25)²]

cost = -√[ - (49/625)]

cost = -√(576/625)

cost = -24 / 25

 

 

Plug in this value of cost into the formula.

 

 

 

 

2)

 

Let      q = 2x

 

 

cos(q) = -√(3)/2

 

q = π - (π/6) = (5π)/3

q = π + (π/6) = (7π)/3

 

 

Substituting back into x.

 

(5π)/3 = 2x                    (7π)/3 = 2x

 

5π / 6 = x                              7π / 6 = x