How to solve trigonometric equations

Solving a trigonometric equation means finding every angle, within a stated range, that makes the equation true. Because $\sin$, $\cos$ and $\tan$ repeat, a single equation such as $\cos 2x=\tfrac{1}{2}$ usually has several solutions in a range like $0^{\circ}\le x\le 360^{\circ}$.

This is a core skill in the Form 5 Trigonometric Functions chapter of Add Math and it carries method marks in both papers. The task is not just to press one calculator button, that gives only a single angle, but to use the basic angle together with the sign of the ratio in each quadrant to list all the correct solutions.

Reach for this method whenever a question sets a trigonometric ratio equal to a value and states a range, using wording such as 'solve $2\cos 2x=1$ for $0^{\circ}\le x\le 360^{\circ}$' or 'find all values of $x$'. The word 'solve' with a range, and the plural 'values', warn you to expect more than one answer.

If the equation contains $\sin^{2}$, $\cos^{2}$ or a mixture of ratios, first use an identity to reduce it to a single ratio; if it contains a multiple angle such as $2x$ or $3x$, remember to widen the range before you solve, then divide back at the end.

1. 1

   Isolate the ratio

   Rearrange until you have a single trig ratio equal to a number, e.g. $\cos 2x=\tfrac{1}{2}$.

2. 2

   Widen the range if needed

   For a multiple angle like $2x$, replace the range $0^{\circ}\le x\le 360^{\circ}$ with $0^{\circ}\le 2x\le 720^{\circ}$.

3. 3

   Find the basic angle

   Take the inverse of the positive value, e.g. $\cos^{-1}(\tfrac{1}{2})=60^{\circ}$. Always use the positive value here.

4. 4

   Decide the quadrants

   The sign of the ratio tells you which quadrants hold solutions, cosine is positive in the 1st and 4th quadrants.

5. 5

   List the angles

   Write every angle for the multiple angle inside the widened range.

6. 6

   Divide back and state

   Divide each angle by the multiple to recover $x$, then list all solutions in the required range.

• Giving only the calculator's first angle and missing the other quadrant.
• Forgetting to widen the range for a multiple angle, for $2x$ the range must run to $720^{\circ}$, not $360^{\circ}$.
• Dividing the range but not every angle, so the extra solutions from the second revolution are lost.
• Getting the quadrants wrong by ignoring whether the ratio is positive or negative.
• Mixing degrees and radians, or rounding the basic angle before the final step.

The step students lose marks on is the range: with a multiple angle like $2x$, they solve inside $0^{\circ}$ to $360^{\circ}$ and hand in only half the answers. In a one-to-one lesson our teachers stop you at that exact line, so widening the range and checking the quadrant signs becomes an automatic habit rather than an afterthought.

Working through the basic angle out loud makes the pattern stick. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we teach trigonometric equations, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.