Solving Trigonometric Equations from 0° to 360°

0:00Let's solve 2 cos theta plus 1 equals 0, for theta from 0 to 360 degrees.

0:08Subtract 1 from both sides, then divide by 2. So cos theta equals negative one half.

0:16On the unit circle, of radius 1, an angle in standard position turns counterclockwise from the positive x-axis.

0:25Where its terminal arm meets the circle, the x-coordinate is cos theta.

0:30We need x to be negative one half, and that line meets the circle twice: two answers.

0:37The reference angle is the acute angle between the terminal arm and the x-axis.

0:43Its cosine is one half over 1, and cos 60 degrees is one half, so the reference angle is 60 degrees.

0:52Cosine is negative where x is negative, in quadrants two and three.

0:58In quadrant two, theta is 180 minus 60, which is 120 degrees.

1:05In quadrant three, theta is 180 plus 60, which is 240 degrees.

1:12Check: both cosines are negative one half, and 2 times negative one half, plus 1, is 0.

1:20So theta is 120 or 240 degrees. Not 60 degrees: that is only the reference angle.

1:29Now sin theta equals 0.6. Inverse sine on a calculator gives only 36.9 degrees. Stopping there is a common mistake.

1:40Sine is the y-coordinate, positive in quadrants one and two, so the other answer is 180 minus 36.9, or 143.1 degrees.

1:53To recap: find the reference angle, use the sign to choose the quadrants, and give every angle from 0 to 360 degrees.