Mathematics.

trigonometric identities

Solving Trigonometric Equations

Trigonometry30 minDifficulty4 out of 10

You should know: trigonometric identities

Overview

A trigonometric equation is an equation involving one or more trigonometric functions of an unknown angle, such as 2sinx − 1 = 0. Because trigonometric functions are periodic, such equations typically have infinitely many solutions, which are usually reported either as a full set differing by multiples of the period or restricted to one period such as [0°, 360°). Solving them combines inverse trigonometric functions (to find a first solution), reference angles and symmetry (to find all solutions in a period), and algebraic techniques like factoring or identity substitution when the equation involves more than one trig function.

Intuition

Because trigonometric functions repeat every period, if x₀ is one solution to an equation like sinx = k, then x₀ plus any whole number of periods is also a solution — the graph literally repeats itself, so wherever the curve crosses the target height once, it crosses it again one period later. Within a single period there are usually two crossing points (except at the extreme values ±1), which is why sinx = k typically has two families of solutions per period, related by the symmetry x ↔ π − x.

Formal Definition

Definition

For the basic equation sin x = k with −1 ≤ k ≤ 1, all solutions are given by:

x=arcsin⁡(k)+2πnorx=π−arcsin⁡(k)+2πn,n∈Zx = \arcsin(k) + 2\pi n \quad \text{or} \quad x = \pi - \arcsin(k) + 2\pi n, \quad n \in \mathbb{Z}
General solution for sin x = k
x=±arccos⁡(k)+2πn,n∈Zx = \pm\arccos(k) + 2\pi n, \quad n \in \mathbb{Z}
General solution for cos x = k
x=arctan⁡(k)+πn,n∈Zx = \arctan(k) + \pi n, \quad n \in \mathbb{Z}
General solution for tan x = k

Worked Examples

  1. Isolate sin x.

    sin⁡x=12\sin x = \frac{1}{2}

  2. The reference angle is 30°; sine is positive in quadrants I and II.

    x=30°orx=180°−30°=150°x = 30° \quad \text{or} \quad x = 180° - 30° = 150°

Answer: x = 30° or x = 150°

Practice Problems

Difficulty 3/10

Solve tan x = 1 for x in [0°, 360°).

Difficulty 5/10

Solve 2sin²x + sinx − 1 = 0 for x in [0°, 360°).

Difficulty 6/10

A projectile's range is R = (v²/g)sin(2θ). For v²/g = 40 m and a target range of R = 20 m, solve for the launch angle θ in (0°, 90°).

Quiz

All solutions of sin x = 1/2 within [0°, 360°) are:
The general solution family for cos x = k is:
When an equation contains sin²x and sinx (e.g. 2sin²x + sinx − 1 = 0), the standard technique is to:

Summary

  • Trigonometric equations generally have infinitely many solutions due to periodicity; solve within one period and add multiples of the period for the general solution.
  • Reference angles plus knowledge of which quadrants make each function positive/negative give all solutions within a period.
  • Equations involving sin²x or cos²x alongside sinx/cosx are solved by substitution into an ordinary polynomial equation, then factoring.

References

  1. WebsiteWikipedia — List of trigonometric identities