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Solving Simple Trig Equations

Use inverse trig functions and the unit circle to find every solution within a given range.

Finding All Solutions

sin(θ) = 0.5 has more than one solution between 0 ° and 360 °, because the sine wave crosses 0.5 twice. A calculator's inverse function only gives one of them — you must use symmetry to find the rest.

Example 1 — sin equation with two solutions

Solve sin(θ) = 0.5 for 0 ° \leq θ \leq 360 °. 1 The calculator gives one solution: θ = sin⁻¹(0.5) = 30 °. 2 Sine is also positive in Quadrant 2 . The second solution is 180 ° - 30 ° = 150 °. 3 Check both are in range: 30 ° and 150 ° are both between 0 ° and 360 °. Answer: θ = 30 ° or θ = 150 °

Example 2 — cos equation with two solutions

Solve cos(θ) = 0.5 for 0 ° \leq θ \leq 360 °. 1 The calculator gives one solution: θ = cos⁻¹(0.5) = 60 °. 2 Cosine is also positive in Quadrant 4 . The second solution is 360 ° - 60 ° = 300 °. Answer: θ = 60 ° or θ = 300 °

Finding the second solution in 0°–360°
Finding the second solution in 0°–360°

This table covers positive values only, matching the examples above. Negative-value cases build on the same ASTC idea.

Practice

Solve sin(θ) = 0.5 for 0 ° \leq θ \leq 360 °. Enter the larger solution.

Your answer

Finding All Solutions

sin(θ) = 0.5 has more than one solution between 0 ° and 360 °, because the sine wave crosses 0.5 twice. A calculator's inverse function only gives one of them — you must use symmetry to find the rest.

Example 1 — sin equation with two solutions

Solve sin(θ) = 0.5 for 0 ° \leq θ \leq 360 °. 1 The calculator gives one solution: θ = sin⁻¹(0.5) = 30 °. 2 Sine is also positive in Quadrant 2 . The second solution is 180 ° - 30 ° = 150 °. 3 Check both are in range: 30 ° and 150 ° are both between 0 ° and 360 °. Answer: θ = 30 ° or θ = 150 °

Example 2 — cos equation with two solutions

Solve cos(θ) = 0.5 for 0 ° \leq θ \leq 360 °. 1 The calculator gives one solution: θ = cos⁻¹(0.5) = 60 °. 2 Cosine is also positive in Quadrant 4 . The second solution is 360 ° - 60 ° = 300 °. Answer: θ = 60 ° or θ = 300 °

Finding the second solution in 0°–360°
Finding the second solution in 0°–360°

This table covers positive values only, matching the examples above. Negative-value cases build on the same ASTC idea.

Practice

Solve sin(θ) = 0.5 for 0 ° \leq θ \leq 360 °. Enter the larger solution.

Your answer