Solving a Simple Trigonometric Equation

A trig equation asks which angles produce a certain ratio. Since sine and cosine repeat every full turn, and since most heights or horizontal positions on the unit circle occur at two different angles within one turn, a single equation almost always has more than one solution. Solving means isolating the trig function first, exactly like isolating x in regular algebra, then asking the unit circle which angles give that value.

Picture the unit circle. sin(x) is the y-coordinate of the point at angle x. Any horizontal line drawn across the circle, except at the very top or bottom, crosses the circle twice, so two angles share that y-value: one in the first or second quadrant pairing, reflected across the vertical axis. If sin(x) = k gives a first-quadrant angle x1, the second solution is π - x1 (or 180 - x1 in degrees). Cosine works the same way but reflects across the horizontal axis instead, giving x1 and -x1 (or 2pi - x1). Once you have the solutions in one cycle, add multiples of 2pi, or 360 degrees, to list every solution on the whole number line.

This sits right after the unit circle definitions of sine and cosine, and right before the two triangle laws, which solve for unknown sides and angles in triangles that are not right triangles. The equation-solving skill here is really the unit-circle picture run backward: instead of being given an angle and asked for a ratio, you are given a ratio and asked for every angle that produces it.