# Suppose A is the arc on the unit circle that connects (1,0) to (-12/13, 5/13) and passes through (0,-1). Let u be the measure of the angle that subtends this arc. Find the value of sin(u)?

Feb 26, 2018

See below.

#### Explanation:

If we pass through point $\left(0 , - 1\right)$ then we are rotating in a anti clockwise direction.

We can see by the coordinates $\left(- \frac{12}{13} , \frac{5}{13}\right)$ that the terminal side is in the II quadrant. The $y$ coordinate $\frac{5}{13}$ is the sine ratio for the angle that is formed with the negative $x$ axis, which will be $\arcsin \left(\frac{5}{13}\right)$, but we have rotated through the IV and III quadrants, so our angle $\boldsymbol{u}$ will be $- \arcsin \left(\frac{5}{13}\right) - \pi \approx - 3.536383774 \textcolor{w h i t e}{8888}$radians or $- {202.62}^{\circ}$.

I have given the angle as a negative rotation, but in angles subtended by arcs this is usually given as a positive angle .i.e. $3.53638377$ radians.

$\sin \left(u\right) = \sin \left(- \arcsin \left(\frac{5}{13}\right) - \pi\right) \approx 0.3846153846$