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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

Why it matters: AC electric current, simple harmonic motion, light waves, and ocean tides are all modeled with trig, and 'when does the value first hit X?' problems require solving trig equations.

Worked example

Problem. Solve sin(x) = 1/2 on [0 degrees, 360 degrees). Enter the smaller solution.

  1. Reference angle: arcsin(1/2) = 30 degrees.
  2. sin is positive in quadrants I and II, so solutions are 30 and 180 - 30 = 150.
  3. The smaller is 30.

Answer: 30

Practice problems

1. sin(x) = 0 on [0, 360). Enter the smallest non-negative solution in degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: sin(x) = 0 on [0, 360). Enter the smallest non-negative solution in degrees.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. sin(0) = 0.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

2. sin(x) = 1 on [0, 360). Enter the solution in degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: sin(x) = 1 on [0, 360). Enter the solution in degrees.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. sin reaches 1 only at 90.
  4. Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.

Answer: 90

3. sin(x) = 1/2 on [0, 360). Enter the smaller solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: sin(x) = 1/2 on [0, 360). Enter the smaller solution.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Reference 30; sin positive in Q I and Q II.
  4. Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.

Answer: 30

4. sin(x) = 1/2 on [0, 360). Enter the larger solution.

Show solution
  1. Core Practice: First identify exactly what the question is asking: sin(x) = 1/2 on [0, 360). Enter the larger solution.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Q II: 180 - 30 = 150.
  4. Check the result by substituting or estimating: the response should match 150 and make sense in the original problem.

Answer: 150

5. cos(x) = 0 on [0, 360). Enter the smaller solution.

Show solution
  1. Core Practice: First identify exactly what the question is asking: cos(x) = 0 on [0, 360). Enter the smaller solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. cos = 0 at 90 and 270; smaller is 90.
  4. Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.

Answer: 90

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