Solving Trigonometric Equations
Isolate the trig function, find the reference angle, list all solutions.
About this topic
A trig equation like 2 sin x − 1 = 0 is solved by isolating the function (sin x = 1/2) and then finding all angles with that value, using the unit circle and the inverse function. Because trig functions are periodic, there are infinitely many solutions unless the interval is restricted.
Find the reference angle, decide which quadrants give the right sign, and then add multiples of the period to give a general solution. For sin x = 1/2, the solutions in one turn are π/6 and 5π/6, so the general solution is x = π/6 + 2πn or x = 5π/6 + 2πn for any integer n.
More involved equations can be handled by factoring, such as 2cos²x − cos x = 0 giving cos x (2cos x − 1) = 0, or by using identities to write everything in terms of one function. For example, replace cos²x with 1 − sin²x to get a quadratic in sin x. Equations with multiple angles like sin 2x = 1/2 need the period to be divided afterwards.
Periodic problems such as tides or daylight hours are answered this way. Never divide both sides by a trig function, since you lose solutions; factor instead. Always check that solutions lie in the requested interval, and give radians or degrees as the problem expects.
Key ideas
- Use the unit circle to find every angle
- Add 2πk for general solutions
- Check for extraneous answers
Quick check
Use cases: where Solving Trigonometric Equations helps
- Homework: review the key idea behind Solving Trigonometric Equations before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Solving Trigonometric Equations without notes.
- Visual learning: play with the interactive lab to see how Solving Trigonometric Equations changes when you adjust its inputs.
- Teaching a friend: explain Solving Trigonometric Equations in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Solving Trigonometric Equations?
Isolate the trig function, find the reference angle, list all solutions.
What will I learn in Solving Trigonometric Equations?
Use the unit circle to find every angle. Add 2πk for general solutions. Check for extraneous answers.
Is Solving Trigonometric Equations part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Solving Trigonometric Equations lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Solving Trigonometric Equations?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.