Solving Trigonometric Equations

A trigonometric equation is any equation that contains one or more trigonometric functions such as sine, cosine, or tangent applied to an unknown angle. Unlike algebraic equations that typically have a finite number of solutions, trigonometric equations often have infinitely many solutions because trigonometric functions are periodic — they repeat their values at regular intervals. For example, sin(θ) = 0 is satisfied not just by θ = 0, but by every integer multiple of π radians. Solving a trigonometric equation means finding all angle values (usually in radians or degrees) that make the equation true. The process involves isolating the trigonometric function, finding a principal (reference) solution using inverse trigonometric functions or known exact values, and then writing the general solution that captures every valid angle using the period of the function. Solutions can be restricted to a specific interval such as [0, 2π) or expressed as an infinite family using integer parameters. Trigonometric equations arise naturally in physics (wave motion, oscillations, optics), engineering (signal processing, mechanical design), and pure mathematics. Mastering their solution requires a solid understanding of the unit circle, reference angles, trigonometric identities, and the behavior of each trigonometric function across its full domain.

Key takeaways
  • Simplify and Isolate
  • Find the Principal Solution(s)
  • Determine All Solutions in One Period
  • Write the General Solution
  • Verify Your Solutions

Simplify and Isolate

Rearrange the equation algebraically so that a single trigonometric function stands alone on one side. Use identities such as the Pythagorean identity sin²θ + cos²θ = 1 or double-angle formulas to reduce the equation to a simpler form. For example, 2sin²θ − 1 = 0 becomes sin²θ = 1/2, so sinθ = ±1/√2.

Find the Principal Solution(s)

Use the inverse trigonometric function to find the principal value in the standard range: arcsin maps to [−π/2, π/2], arccos to [0, π], arctan to (−π/2, π/2). For exact values, refer to the unit circle: sinθ = 1/2 gives θ = π/6 as the principal value. For non-exact values, use a calculator in radian mode.

Determine All Solutions in One Period

Because sine and cosine complete one full cycle over [0, 2π), use symmetry of the unit circle to find all solutions within that interval. For sinθ = k where k > 0, there are solutions in quadrants I and II: θ = α and θ = π − α, where α is the reference angle. For cosθ = k, solutions appear in quadrants I and IV: θ = α and θ = 2π − α. For tanθ = k, there is one solution per period of π: θ = α and θ = π + α.

Write the General Solution

Extend the solutions to cover all real angles by adding integer multiples of the period. The period of sin and cos is 2π, so the general solution for sinθ = k is θ = α + 2nπ or θ = (π − α) + 2nπ, where n ∈ ℤ. The period of tan is π, so the general solution for tanθ = k is θ = α + nπ, where n ∈ ℤ. If the problem restricts solutions to an interval like [0, 2π), substitute integer values of n to list only those solutions.

Verify Your Solutions

Substitute each solution back into the original equation to confirm it is valid. This step is especially important when the equation was squared during simplification, since squaring can introduce extraneous solutions. For example, solving sinθ = −cosθ by squaring gives tan²θ = 1, but not every solution of tan²θ = 1 satisfies the original equation.

The Key Equation

θ=arcsin⁡(k)+2nπorθ=(π−arcsin⁡(k))+2nπ,n∈Z\theta = \arcsin(k) + 2n\pi \quad \text{or} \quad \theta = (\pi - \arcsin(k)) + 2n\pi, \quad n \in \mathbb{Z}
  • \theta — Angle (unknown) (radians): The angle whose sine equals k; the variable being solved for.
  • k — Constant (dimensionless): The value that sin θ equals; must satisfy −1 ≤ k ≤ 1 for real solutions to exist.
  • \arcsin(k) — Principal value (radians): The unique angle in [−π/2, π/2] whose sine is k, obtained from the inverse sine function.
  • n — Integer parameter (dimensionless): Any integer (…, −2, −1, 0, 1, 2, …) that generates successive solutions by adding full periods.
  • 2\pi — Period of sine (radians): The length of one complete cycle of the sine function; solutions repeat every 2π radians.

Worked Example

Problem. Solve 2cos²θ − cosθ − 1 = 0 for θ ∈ [0, 2π).

θ=0,  2π3,  4π3\theta = 0,\; \dfrac{2\pi}{3},\; \dfrac{4\pi}{3}

Answer. θ = 0, 2π/3, 4π/3

Figure
Diagram
xyOθ=π/24π/35π/302π/3π/3y=1/2Unit circle: solutions of cos θ = −1/2 (purple) and sin θ = 1/2 (orange line)
Origins · 1 minLeonhard Euler and how this was discovered
LE
Leonhard Euler
1707–1783 · Swiss

Euler standardized the modern notation for trigonometric functions and developed the famous formula e^(iθ) = cosθ + i sinθ, which unifies trigonometry with complex exponentials. This relationship, known as Euler's formula, is the theoretical foundation behind expressing trigonometric equations in exponential form and solving them with complex analysis. He also introduced the radian measure systematically and catalogued solutions to many classes of trigonometric equations. Euler lost sight in his right eye around age 31 and became nearly totally blind by age 59, yet he produced roughly half of his total mathematical output after losing most of his vision, largely by dictating to assistants.

  • Unit Circle and Radian Measure — The unit circle provides the geometric foundation for understanding why trigonometric equations have multiple solutions and how to locate all solutions using reference angles.
  • Inverse Trigonometric Functions — Arcsin, arccos, and arctan are the tools used to extract the principal solution from a trigonometric equation before writing the general solution.
  • Trigonometric Identities — Identities such as the Pythagorean identity and double-angle formulas are used to simplify complex trigonometric equations into solvable forms.
  • Graphs of Trigonometric Functions — Visualizing the graph of y = sinθ or y = cosθ helps confirm the number and location of solutions, and shows clearly how periodicity creates infinitely many answers.
  • Solving Quadratic Equations — Many trigonometric equations, after substitution, become quadratic equations in a trig function. Factoring or the quadratic formula is then applied exactly as in algebra.