Question

How do you evaluate `Sin^2 (pi/9) + Cos^2( pi/9)`?

Answer 1

`1`

Explanation:

Without even considering the arguments of sine and cosine, there is an identity that for all `x`, `sin^2(x) + cos^2(x) = 1`.

One way of seeing that this is true is to consider the unit circle, and note that for any point on the circle with angle `theta`, `cos(theta)` is the `x`-coordinate of that point, and `sin(theta)` is the `y`-coordinate. Then, drawing a right triangle with the hypotenuse connecting the origin and the point, we find that the triangle has legs of length `cos(theta)` and `sin(theta)`, and a hypotenuse of length `1`. Applying the Pythagorean theorem gives us the desired result.