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Math101learn.math101.caSolving Trigonometric Equations
Trigonometric equations combine algebra, reference angles, periodicity, and domain restrictions to find every valid angle.
A trig equation rarely has only one answer: periodic functions repeat, so the interval is part of the problem.
A reliable process
- State the interval and angle unit.
- Rearrange or factor to isolate a trig function.
- Find a reference angle or exact benchmark.
- Use signs to choose quadrants.
- list every solution in the interval.
- Check the original equation and restrictions.
Calculator mode must match degrees or radians.
Basic equation in degrees
Solve
Then $\sin x=1/2$. The reference angle is $30^\circ$, and sine is positive in Quadrants I and II. Therefore
The interval excludes $360^\circ$, though it would not be a solution here anyway.
Worked example: factor a trig quadratic
General solutions
Without a restricted interval, express periodic families. In degrees:
has
or
Cosine has symmetry $x=\pm\alpha+360^\circ k$, and tangent repeats every $180^\circ$.
Radian solutions
The method is unchanged in radians. Solve
The reference angle is $\pi/4$, and sine is negative in Quadrants III and IV:
Use exact radian values unless a decimal is requested.
Compound angles
For
the compound angle $2x$ ranges from $0^\circ$ to $720^\circ$. Solve for all $2x$ values across two cycles, then divide each by $2$.
Failing to expand the compound-angle interval is a common way to lose half the solutions.
Using identities
Convert multiple trig functions to a common function. The identity
can create a quadratic in $\cos x$. Double-angle and sum identities may also simplify an equation.
Identity substitutions must preserve the original domain.
Equations with denominators
Record values that make any original denominator zero. After multiplying through or cancelling, reject excluded angles even if they satisfy the transformed equation.
Dividing by a trig expression can also lose solutions when that expression might be zero. Factoring and using the zero-product property is often safer.
Numerical equations
An equation such as $\sin x=0.37$ may require inverse sine for a reference value, followed by quadrant reasoning. More complex mixed equations may need graphing or numerical root-finding.
Use a window covering the full requested interval and verify every approximate root in the original equation.
Context and units
Sinusoidal models may ask for times when a height, temperature, or tide reaches a threshold. Restrict solutions to the model's time domain and interpret repeated answers with units.
An algebraic angle solution is not complete until it is translated back to the contextual input.
Common mistakes
Giving only the calculator's principal angle. Use reference angles and periodicity.
Mixing degrees and radians. Match the interval and calculator mode.
Dividing by a trig factor that could be zero. This can discard solutions.
Solving $2x$ over the interval for $x$. Adjust the compound-angle interval first.
Keeping excluded denominator values. Check the original equation.
Quick self-check
- What is the exact interval and angle unit?
- Is the equation isolated, factored, or converted to one trig function?
- What is the reference angle and which quadrants apply?
- Have all cycles in the interval been considered?
- Were any solutions lost through division or introduced through transformation?
- Do all final values verify in the original equation?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve 2 sin x − 1 = 0 for 0° ≤ x < 360°.
- sin x = 1/2.
- The reference angle is 30°.
- The interval solutions are 30° and 150°.
End of lesson
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