Expert: Azeem Hussain - 3/15/2009

Question
the question is: Find all the angles (A) (between 0 and 2pi) that sastisfy the equation sineA = -1

i used the inverse sine and found A = -1.5708. but are there multiple answers? and how can i convert my answer found to be between 0 and 2pi?

thanks so much if u can help

Answer
Hi Olivia!

The easiest way to visualize this is to draw a circle centred at the origin with a radius of 1.  sin A corresponds to the y-value.  By observation, there is only one point where the y-value is -1: on the y-axis joining quadrants 3 and 4.

The angle is measured from the line segment in the first quadrant from (0,0) to (1,0) (3 o'clock if you'd like to think of it like that) to the point(s) in question.  Remember, there are 2π radians in one revolution.  Your point is 3/4 of one revolution so 3/4(2π)=3π/2.  This is between 0 and 2π.  (Measuring the angle from the negative will give -π/2, but that does not land within the desired interval.)

With exception of the extremities which will give one answer (either 1 or -1), the sine or cosine of any other angle will have two values between 0 and 2π.  For example, there are two places where sin A=0.5: at π/6 in quadrant 1 and at 5π/6 in quadrant 2.

Thanks for asking,
Azeem