A trigonometric equation rarely has only one solution. Sine, cosine and tangent repeat, so the stated interval is part of the mathematics: it tells you which members of a repeating solution family must be reported.
1. Isolate the trigonometric function
Solve 2sin x=1 for 0°≤x≤360°. First write sin x=1/2. The reference angle is 30°.
Sine is positive in quadrants I and II, so x=30° or 150°.
2. Use the interval, not only the inverse key
sin⁻¹(1/2)=30° gives a principal value. It does not list every solution in 0°≤x≤360°. The unit-circle signs and interval provide the remaining solution.
3. Cosine example
Solve cos x=−√3/2 for 0≤x≤2π. The reference angle is π/6. Cosine is negative in quadrants II and III.
Therefore x=5π/6 or 7π/6.
4. Tangent repeats every π
Solve tan x=1 for −π≤x≤π. A principal solution is π/4. Tangent has period π, so nearby solutions are π/4+kπ.
Within the interval, x=−3π/4 or π/4.
5. Input multipliers change the interval for the angle
Solve sin(2x)=1/2 for 0°≤x≤180°. Let θ=2x. Then 0°≤θ≤360°.
θ=30° or 150°, so x=15° or 75°.
Transform the interval together with the variable. Solving for θ on the original x-interval would miss or add solutions.
6. Shifted input example
Solve cos(x−30°)=1/2 for 0°≤x≤360°. Let θ=x−30°, so −30°≤θ≤330°.
Cos θ=1/2 gives θ=60° or 300° in that interval, hence x=90° or 330°.
7. Endpoint checks matter
For an inclusive interval, test whether endpoints satisfy the equation. For sin x=0 on 0≤x≤2π, the solutions are 0, π and 2π.
The endpoints represent different allowed input values even though sine has the same output at both.
8. Degrees and radians must remain consistent
A radian interval such as 0≤x≤2π requires radian reasoning or a complete consistent conversion. Calculator mode should match the angle unit.
9. Check solutions in the original equation
Substitution is especially useful after algebraic manipulation or an input transformation. It confirms both the trigonometric value and the interval condition.
10. Common mistakes
- Reporting only the calculator’s principal inverse value.
- Forgetting the transformed interval for 2x, 3x or x−a.
- Mixing degrees and radians.
- Missing inclusive endpoints.
- Using sine quadrant signs for cosine or tangent.
11. Practice
- Solve sin x=√3/2 for 0°≤x≤360°.
- Solve cos x=−1/2 for 0≤x≤2π.
- Solve tan x=−1 for 0°≤x<360°.
- Solve sin(2x)=0 for 0°≤x≤180°.
- Solve cos(x+π/4)=0 for 0≤x≤2π.
12. Answers
- 60°,120°.
- 2π/3,4π/3.
- 135°,315°.
- 2x=0°,180°,360°, so x=0°,90°,180°.
- Let θ=x+π/4, so π/4≤θ≤9π/4. cosθ=0 at π/2,3π/2,5π/2; the last is outside. Hence x=π/4,5π/4.
Continue with the Trigonometric Equations guide, Trigonometric Functions and Graphs, or return to the Additional Mathematics Hub.