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Trigonometric Equation — Definition, Formula & Examples

A trigonometric equation is an equation that contains a trig function such as sine, cosine, or tangent of an unknown angle. It is true only for certain angles, and your job is to find them.

A trigonometric equation is a conditional equation in which the unknown appears as the argument of at least one trigonometric function. Its solution set is the set of all values in the domain that make it true. Trig functions are periodic, so an equation with no interval restriction usually has infinitely many solutions. You write these as families, for example x=30∘+360∘nx = 30^\circ + 360^\circ n with nn an integer. This sets it apart from a trigonometric identity, which holds for every value where both sides are defined.

Key Formula

sin⁡x=k  ⟹  x=α+360∘norx=(180∘−α)+360∘n\sin x = k \;\Longrightarrow\; x = \alpha + 360^\circ n \quad\text{or}\quad x = (180^\circ - \alpha) + 360^\circ n
Where:
  • kk = A constant with −1≤k≤1-1 \le k \le 1
  • α\alpha = The principal value sin⁡−1(k)\sin^{-1}(k), given by a calculator
  • nn = Any integer, which accounts for every coterminal angle

Worked Example

Problem: Solve 2sin⁡x−1=02\sin x - 1 = 0 for 0∘≤x<360∘0^\circ \le x < 360^\circ.
Step 1: Isolate the trig function. Add 1 to both sides, then divide by 2.
sin⁡x=12\sin x = \frac{1}{2}
Step 2: Find the reference angle. It is the acute angle whose sine is 12\tfrac{1}{2}.
α=sin⁡−1 ⁣(12)=30∘\alpha = \sin^{-1}\!\left(\tfrac{1}{2}\right) = 30^\circ
Step 3: Sine is positive in Quadrants I and II. The Quadrant II angle with a 30∘30^\circ reference angle is 180∘−30∘180^\circ - 30^\circ.
x=30∘orx=180∘−30∘=150∘x = 30^\circ \quad\text{or}\quad x = 180^\circ - 30^\circ = 150^\circ
Answer: x=30∘x = 30^\circ or x=150∘x = 150^\circ

Why It Matters

Trig equations take up much of a precalculus course. They return in calculus when you set a derivative such as cos⁡x\cos x equal to zero to locate maxima and minima. Physics and engineering students solve them to find when a wave, a pendulum, or an AC voltage reaches a given value.

Common Mistakes

Mistake: Stopping at the single calculator answer, such as x=30∘x = 30^\circ, and missing 150∘150^\circ.
Correction: The inverse sine key returns only one angle. Check which quadrants give the correct sign, then list every angle in the required interval. With no interval given, add 360∘n360^\circ n to each solution.

Related Terms