General Solutions of Trigonometric Equations

General Solutions of Trigonometric Equations

Trigonometric equations usually have repeating solutions. Find the reference angle, use the sign/quadrant to identify solutions in the specified interval, then use the period for a general solution if requested. In degrees, sine and cosine repeat every $360^\circ$; tangent repeats every $180^\circ$.

Worked reasoning

Solve $\sin\theta=1/2$ for $0^\circ\leq\theta\leq360^\circ$. Reference angle is $30^\circ$; sine is positive in I and II, giving $30^\circ$ and $150^\circ$.

Exam method

1. Isolate one trig ratio. 2. Find the reference angle. 3. Use quadrants and the requested interval/period to list all solutions.

Interval notation is mathematical communication

Before solving, circle the interval or domain in the question. A solution at $360^\circ$ may be included in $0^\circ\leq\theta\leq360^\circ$ but excluded in a half-open interval. For tangent, add multiples of $180^\circ$; for sine/cosine, add multiples of $360^\circ$ when writing general solutions. List solutions in increasing order and check each by the sign of the relevant ratio, not only the reference angle.

One-minute retrieval: General Solutions of Trigonometric Equations

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.