Solving Trigonometric Equations
A free Algebra II lesson from the “Trigonometry and Modeling” unit, with a worked example and practice problems including step-by-step solutions.
Trig equations like sin(x) = 1/2 have many solutions because sine repeats every 360 degrees. On [0, 360) most equations sin(x) = k and cos(x) = k have TWO solutions; tan(x) = k has solutions repeating every 180. Use the unit circle: find the reference angle, then locate every quadrant that gives the right sign.
What you'll learn
- Solve simple equations of the form sin(x) = k, cos(x) = k, tan(x) = k on a finite interval
- Find ALL solutions on the standard interval [0 degrees, 360 degrees)
- Use the unit circle and reference angles to locate solutions
Worked example
Problem. Solve sin(x) = 1/2 on [0 degrees, 360 degrees). Enter the smaller solution.
- Reference angle: arcsin(1/2) = 30 degrees.
- sin is positive in quadrants I and II, so solutions are 30 and 180 - 30 = 150.
- The smaller is 30.
Answer: 30
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. sin(x) = 0 on [0, 360). Enter the smallest non-negative solution in degrees.
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- sin(0) = 0.
Answer: 0
2. sin(x) = 1 on [0, 360). Enter the solution in degrees.
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- sin reaches 1 only at 90.
Answer: 90
3. sin(x) = 1/2 on [0, 360). Enter the smaller solution.
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- Reference 30; sin positive in Q I and Q II.
Answer: 30
4. sin(x) = 1/2 on [0, 360). Enter the larger solution.
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- Q II: 180 - 30 = 150.
Answer: 150
5. cos(x) = 0 on [0, 360). Enter the smaller solution.
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- cos = 0 at 90 and 270; smaller is 90.
Answer: 90
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