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Math Practice

Solving Trigonometric Equations Practice Problems

Isolate the trig function, find every angle in one turn, and check the quadrants. The later problems need factoring or a wider range for multiple angles.

  1. 1.

    Solve for .

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    : quadrants I and II.

  2. 2.

    Solve for .

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    : quadrants I and IV.

  3. 3.

    Solve for .

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    , and tangent repeats every .

  4. 4.

    Solve for .

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    Sine is zero at 0 and .

  5. 5.

    Solve for .

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    : quadrants II and III, reference angle .

  6. 6.

    Solve for .

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    : quadrants III and IV.

  7. 7.

    Solve for .

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    : gives 0 and ; gives 0 again.

  8. 8.

    Find the smallest positive solution of , in radians to three decimal places.

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    .

  9. 9.

    Solve for .

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    : gives ; gives .

  10. 10.

    Solve for .

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    : reference angle in every quadrant.

  11. 11.

    Solve for .

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    With on : ; halve each.

  12. 12.

    How many solutions does have? Type 0 if there are none.

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    Sine never exceeds 1, so there are no solutions.
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MPhil in Mathematics, Founder of Mathovia
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Solving Trigonometric Equations Practice Problems

These problems follow the solving trigonometric equations lesson. Give every solution in .

The general solutions

Common mistakes to watch for

  • Stopping at one solution. has two solutions in : and .
  • Dividing by a trig function. Dividing by loses the solutions where ; factor instead.
  • Wrong interval for multiple angles. For on , solve over , then halve.
  • Keeping impossible values. has no solution; discard that factor.

Frequently Asked Questions

How do I type several solutions?+

Separate them with commas, for example pi/6, 5pi/6. The order does not matter, and decimals within 0.001 are accepted.

Why give answers only in [0, 2π)?+

Every other solution is one of these plus a multiple of the period, so one turn describes them all.

More practice problems

Inverse Trigonometric Functions Practice Problems Trigonometric Functions Practice Problems Quadratic Equations Practice Problems