Question

If θ is a Quadrant II angle and cosθ = -2/3 , then sinθ = _____.

Answer 1

See the attached picture.  If a point P(x, y) is on the terminal side of an angle theta.

$\cos(\theta) = \frac{x}{r}$

In this case,  x = -2 and r = 3. Use the Pythagorean Theorem to find the value of y.

$x^2 + y^2 =r^2 \\ 4+y^2=9 \\ y^2 = 5 \\ y=\sqrt{5}$

Note that you pick the positive square root for  y  because the point is in Quadrant II.

Now, $\sin(\theta) = \frac{y}{r} = \frac{\sqrt{5}}{3}$