Find the exact value of the expression below if theta equals 30 degrees?

Question

f (θ) = sin θ and g (θ) = cos θ. Find the exact value of the expression below if θ = 30 degrees. Do not use a calculator.

[g(θ)]² =

Answer 1

For this question you have to be familiar with the unit circle. For all values of theta, the unit circle coordinates, (x,y) correspond to (cos(theta), sin(theta)).

The unit circle coordinates for the first quadrant are as follows:

for theta = 0: (1,0)

for theta = 30: (√3/2, 1/2)

for theta = 45: (√2/2, √2/2)

for theta = 60: (1/2, √3/2)

for theta = 90: (0,1)

If you draw this out and reflect over the x and y axes, you can find the remaining coordinates for the additional angle measures, they are all reflections so the signs will change. This is a good exercise to do on your own; you should be able to draw the first quadrant from memory.

Thus, g(theta) = cos(theta) = cos(30) = √3/2. If you square this it's 3/4.