Question

Point b has coordinates (3,-4) and lies on the circle whose equation is x^2 + y^2= 25. If angle is drawn in a standard position with its terminal ray extending through point b, what is the sine of the angle?

Answer 1

Point B has coordinates (3,-4) and lies on the circle. Draw the perpendiculars from point B to the x-axis and y-axis. Denote the points of intersection with x-axis A and with y-axis C. Consider the right triangle ABO (O is the origin), by tha conditions data: AB=4 and AO=3, then by Pythagorean theorem:


`BO^2=AO^2+AB^2 \\ BO^2=3^2+4^2 \\ BO^2=9+16 \\ BO^2=25 \\ BO=5`.

{Note, that BO is a radius of circle and it wasn't necessarily to use Pythagorean theorem to find BO}
The sine of the angle BOA is

`\sin \angle BOA= (AB)/(BO) = (4)/(5) =0.8`

Since point B is placed in the IV quadrant, the sine of the angle that is drawn in a standard position with its terminal ray will be



`\sin \theta=-0.8` .