Solving Standard Trigonometric Equations: General Solutions

When you see an equation like sin x = 1/2, the question is simple: which angles have a sine of one half? Because the sine function repeats its values forever as you travel around the unit circle, a trigonometric equation does not have just one or two answers. It has infinitely many solutions, spread out at regular intervals across the whole number line.

This matters because real waves never stop. A swinging pendulum or an alternating current keeps oscillating, so engineers and physicists need a formula that lists every solution at once. General solution formulas do exactly that: they pack infinitely many answers into a single expression built around an integer n (0, ±1, ±2, …).

Key Concepts

• General solution: a formula using an integer n that generates every angle satisfying the equation.
• n as an integer: the letter n stands for any whole number, positive, negative, or zero.
• Quadratic in a trig function: e.g. 2sin²x − 3sin x + 1 = 0, solved by substituting t = sin x.

Formulas

All angles below are in radians and n is any integer.

• sin x = sin α gives x = nπ + (−1)ⁿα, producing the two mirror solutions α and π − α in each cycle.
• cos x = cos α gives x = 2nπ ± α, since both α and −α share the same cosine.
• tan x = tan α gives x = nπ + α. Tangent repeats every π, so this formula is the simplest of the three.
• Special cases: sin x = 0 gives x = nπ; cos x = 0 gives x = (2n + 1)π/2; tan x = 0 gives x = nπ.

Worked Example

Solve 2sin²x − 3sin x + 1 = 0 for all real x.

1. Substitute t = sin x. The equation becomes the quadratic 2t² − 3t + 1 = 0.
2. Factor it: 2t² − 3t + 1 = (2t − 1)(t − 1).
3. Set each factor to zero: t = 1/2 or t = 1.
4. Case 1: sin x = 1/2. One solution is α = π/6, so x = nπ + (−1)ⁿ(π/6).
5. Case 2: sin x = 1. The basic solution is α = π/2, and the formula gives x = nπ + (−1)ⁿ(π/2), which simplifies to x = π/2 + 2nπ.
6. Verify with x = π/6: sin(π/6) = 1/2, so 2(1/4) − 3(1/2) + 1 = 1/2 − 3/2 + 1 = 0. Correct.

The full solution set: x = nπ + (−1)ⁿ(π/6) or x = π/2 + 2nπ.

Common Misconceptions

• "sin x = 1/2 has only two solutions." It has two inside [0, 2π] (π/6 and 5π/6) but infinitely many overall.
• "Cosine also uses (−1)ⁿ." It does not; cosine uses x = 2nπ ± α.
• "You can divide both sides by sin x." Dividing throws away every solution where sin x = 0. Always factor instead.

Real-World Applications

• Electrical engineering: alternating current and voltage are sine waves; general solutions find every instant when the current crosses a target value.
• Physics: pendulums and vibrating strings are modelled with sine and cosine; solving for time means solving trig equations.
• Signal processing: Fourier analysis breaks sounds and radio signals into sine waves, and engineers solve trig equations to tune filters.

The three general formulas list every solution of a solvable trig equation: x = nπ + (−1)ⁿα for sine, x = 2nπ ± α for cosine, and x = nπ + α for tangent, with n over all integers. Quadratics in sin x or cos x are handled by substituting t and factoring first, as the worked example showed.