Question

How do you evaluate `cos ((15pi)/4)`?

Answer 1

`cos((15pi)/4) = cos(2*(2pi)-pi/4) = cos(-pi/4) = 1/sqrt(2) = sqrt(2)/2~=0.7071`

Explanation:

We can look at this by considering the angle on the unit circle, where `cos` is the projection on the `x`-axis:

graph{(y^2+x^2-1)((y+0.7071)^2+(x-0.7071)^2-.001)((y)^2+(x-0.7071)^2-.001)=0 [-2.434, 2.433, -1.215, 1.217]}

From this it is easy to see that the same value repeats itself for each full revolution on the circle, i.e. `theta = n*2pi`. In other words:

`cos(n*2pi + theta) = cos(theta)`

In our case we have an angle which is almost 4 full rotations:

`(15pi)/4 = (16pi)/4 - pi/4 = 4pi -pi/4 = 2*(2pi)-pi/4`

Therefore

`cos((15pi)/4) = cos(2*(2pi)-pi/4) = cos(-pi/4) = 1/sqrt(2) = sqrt(2)/2~=0.7071`