Solving Trigonometric Equations
Trigonometric equations contain trig functions like sin, cos, or tan set equal to a value. Solving them requires finding all angles that satisfy the equation — both specific solutions in an interval and general solutions for all periods.
This guide covers the step-by-step method, factoring techniques, identity substitution, quadratic trig forms, multiple-angle equations, worked examples, and a practice quiz.
1Introduction
Unlike algebraic equations where each equation typically has a finite number of solutions, trigonometric equations can have infinitely many solutions because trig functions are periodic. The equation sin x = 1/2, for example, has two solutions per cycle — and the cycle repeats forever.
The key challenge is finding all solutions systematically. We use the unit circle, the ASTC rule (All Students Take Calculus) for determining which quadrants give positive values, and algebra techniques like factoring and substitution.
Isolate the trig function, find the reference angle, determine which quadrants the solutions lie in using ASTC, then write the general solution by adding the period.
Trig equations arise in physics (finding when a projectile reaches a certain height), engineering (AC circuit analysis), navigation (bearing calculations), and advanced math (calculus integrals, differential equations).
2Key Definitions
Trigonometric Equation
An equation involving trig functions of an unknown angle, e.g., 2sin x − 1 = 0.
Principal Solution
Solutions within one period, typically [0, 2π) or [0°, 360°).
General Solution
All solutions, written as principal solutions plus multiples of the period: x = α + 2πn.
Reference Angle
The acute angle between the terminal side and the x-axis. Used to find solutions in all quadrants.
ASTC Rule
“All Students Take Calculus” — tells which trig functions are positive in each quadrant: All (QI), Sin (QII), Tan (QIII), Cos (QIV).
Period
The interval after which the function repeats. sin/cos: 2π. tan: π.
Extraneous Solution
A “solution” that appears during solving but does not satisfy the original equation. Always check!
Quadratic in Trig Form
An equation like 2cos²x − 3cos x + 1 = 0 that can be solved by substituting u = cos x.
3Step-by-Step Method
Follow this systematic approach for solving any trig equation:
- Isolate the trig function. Use algebra to get a single trig function on one side: sin x = k, cos x = k, or tan x = k.
- Check the value. Is k within the range? sin/cos: [−1, 1]. tan: all reals. If not, no solution.
- Find the reference angle. Use inverse trig: α = arcsin|k|, arccos|k|, or arctan|k|.
- Determine quadrants. Use the ASTC rule and the sign of k to identify which quadrants contain solutions.
- Write principal solutions. Use the reference angle and quadrant rules to find solutions in [0, 2π) or [0°, 360°).
- Write general solutions. Add + 2πn (for sin/cos) or + πn (for tan), where n is any integer.
Quadrant Formulas for Reference Angle α
| Quadrant | Angle (radians) | Angle (degrees) |
|---|---|---|
| QI | α | α |
| QII | π − α | 180° − α |
| QIII | π + α | 180° + α |
| QIV | 2π − α | 360° − α |
ASTC Quick Reference
QII: S
Only sin (+)
QI: A
All positive
QIII: T
Only tan (+)
QIV: C
Only cos (+)
4Solving with Factoring
When a trig equation has multiple terms, factoring is often the best approach. Set the equation equal to zero, factor, then set each factor equal to zero.
Example: 2sin x cos x − cos x = 0
Step 1: Factor out cos x: cos x(2sin x − 1) = 0
Step 2: Set each factor to zero:
cos x = 0 ⇒ x = π/2, 3π/2
2sin x − 1 = 0 ⇒ sin x = 1/2 ⇒ x = π/6, 5π/6
Solutions in [0, 2π): π/6, π/2, 5π/6, 3π/2
Never divide both sides by a trig function. Dividing by cos x in the example above would lose the solutions where cos x = 0 (namely π/2 and 3π/2). Always factor instead of dividing.
5Using Identities
When an equation involves more than one trig function, use identities to convert everything to a single function. The most common strategies:
Pythagorean Identity
sin²x + cos²x = 1. Replace sin²x with 1 − cos²x (or vice versa) to get one trig function.
Double-Angle Identities
sin(2x) = 2sin x cos x. cos(2x) = 2cos²x − 1 = 1 − 2sin²x. Useful for matching angles.
Tangent = sin/cos
Rewrite tan in terms of sin and cos when the equation mixes them.
Example: 2sin²x + cos x − 1 = 0
Step 1: Replace sin²x with 1 − cos²x:
2(1 − cos²x) + cos x − 1 = 0
Step 2: Expand and simplify:
2 − 2cos²x + cos x − 1 = 0
−2cos²x + cos x + 1 = 0
2cos²x − cos x − 1 = 0
Step 3: Factor: (2cos x + 1)(cos x − 1) = 0
Step 4: Solve each factor:
cos x = −1/2 ⇒ x = 2π/3, 4π/3
cos x = 1 ⇒ x = 0
Solutions in [0, 2π): 0, 2π/3, 4π/3
When deciding which identity to use, look at which trig function appears most. Convert everything to that function. If you see sin² and cos, convert sin² to 1 − cos² to get a quadratic in cos.
6Quadratic Forms
Some trig equations have the form a(trig)² + b(trig) + c = 0. Treat the trig function as a variable and use quadratic techniques: factoring, completing the square, or the quadratic formula.
Quadratic Formula for Trig
For au² + bu + c = 0 where u = sin x, cos x, or tan x:
u = (−b ± √(b² − 4ac)) / 2a
After finding u, remember to check: is u within the range of the trig function?
Example: 2cos²x − 3cos x + 1 = 0
Step 1: Let u = cos x. Equation becomes: 2u² − 3u + 1 = 0.
Step 2: Factor: (2u − 1)(u − 1) = 0.
Step 3: u = 1/2 or u = 1.
Step 4: Back-substitute:
cos x = 1/2 ⇒ x = π/3, 5π/3
cos x = 1 ⇒ x = 0
Solutions in [0, 2π): 0, π/3, 5π/3
Example: tan²x − 3 = 0
Step 1: tan²x = 3, so tan x = ±√3.
Step 2: tan x = √3 ⇒ x = π/3, 4π/3 (QI, QIII).
Step 3: tan x = −√3 ⇒ x = 2π/3, 5π/3 (QII, QIV).
Solutions in [0, 2π): π/3, 2π/3, 4π/3, 5π/3
General solution: x = π/3 + πn/3 (or equivalently π/3 + nπ/3 for n = 0, 1, 2, ...)
When taking the square root of both sides (e.g., cos²x = 1/2), always include both ± roots. Missing the negative root means losing half the solutions.
7Equations with Multiple Angles
Equations like sin(2x) = 1/2 or cos(3x) = 0 involve a multiple angle (2x, 3x, etc.). The key technique: solve for the multiple angle first, then divide.
Method for Multiple-Angle Equations
- Let u = 2x (or 3x, etc.). Solve the equation for u in a scaled interval.
- For sin(2x) with x ∈ [0, 2π), solve for 2x ∈ [0, 4π) — double the interval.
- Find all solutions for u in the scaled interval.
- Divide each solution by 2 (or 3, etc.) to get x.
- For general solutions, divide the period addition too: 2x = α + 2πn becomes x = α/2 + πn.
Example: sin(2x) = √3/2 in [0, 2π)
Step 1: Let u = 2x. Solve sin u = √3/2 in [0, 4π).
Step 2: Reference angle = π/3. sin is positive in QI and QII.
Step 3: In [0, 4π): u = π/3, 2π/3, π/3 + 2π, 2π/3 + 2π = π/3, 2π/3, 7π/3, 8π/3.
Step 4: Divide by 2: x = π/6, π/3, 7π/6, 4π/3.
Solutions: π/6, π/3, 7π/6, 4π/3
Example: tan(3x) = 1, general solution
Step 1: Let u = 3x. tan u = 1 ⇒ u = π/4 + πn.
Step 2: Divide by 3: x = π/12 + πn/3.
General solution: x = π/12 + πn/3, n ∈ ℤ
When solving sin(2x) in [0, 2π), students often forget to double the interval for the substitution. If x ∈ [0, 2π), then 2x ∈ [0, 4π), which means you need two full cycles of solutions for 2x.
8Worked Examples
Example 1: Solve 2cos x + 1 = 0 in [0, 2π)
Step 1: Isolate: cos x = −1/2.
Step 2: Reference angle: arccos(1/2) = π/3.
Step 3: cos is negative in QII and QIII.
Step 4: QII: π − π/3 = 2π/3. QIII: π + π/3 = 4π/3.
Answer: x = 2π/3, 4π/3
Example 2: Solve sin²x − sin x = 0 in [0, 2π)
Step 1: Factor: sin x(sin x − 1) = 0.
Step 2: sin x = 0 ⇒ x = 0, π.
Step 3: sin x = 1 ⇒ x = π/2.
Answer: x = 0, π/2, π
Example 3: Solve 2sin²x − 1 = 0, general solution
Step 1: sin²x = 1/2 ⇒ sin x = ±√2/2.
Step 2: sin x = √2/2 ⇒ x = π/4, 3π/4 (QI, QII).
Step 3: sin x = −√2/2 ⇒ x = 5π/4, 7π/4 (QIII, QIV).
General: x = π/4 + πn/2 (equivalently π/4 + nπ/2)
Example 4: Solve cos(2x) = cos x in [0, 2π)
Step 1: Use double-angle: 2cos²x − 1 = cos x.
Step 2: Rearrange: 2cos²x − cos x − 1 = 0.
Step 3: Factor: (2cos x + 1)(cos x − 1) = 0.
Step 4: cos x = −1/2 ⇒ x = 2π/3, 4π/3.
Step 5: cos x = 1 ⇒ x = 0.
Answer: x = 0, 2π/3, 4π/3
Example 5: Solve 3tan²x − 1 = 0 in [0°, 360°)
Step 1: tan²x = 1/3 ⇒ tan x = ±1/√3 = ±√3/3.
Step 2: Reference angle = 30°.
Step 3: tan x = √3/3 (positive): QI = 30°, QIII = 210°.
Step 4: tan x = −√3/3 (negative): QII = 150°, QIV = 330°.
Answer: 30°, 150°, 210°, 330°
9Memory Aids
QI: All positive. QII: Sine positive. QIII: Tangent positive. QIV: Cosine positive. This tells you which quadrants to use for each sign.
When you see a common trig factor, factor it out instead of dividing. Dividing by sin x or cos x loses solutions where they equal zero.
Since tan has period π, its general solutions use + πn. Since sin and cos have period 2π, they use + 2πn. Never mix them up.
For sin(kx) with x ∈ [0, 2π), solve for kx ∈ [0, 2kπ). Double angle = double interval. Triple angle = triple interval.
After solving, if you get sin x = 2 or cos x = −3, stop — those have no solution. sin and cos are bounded by [−1, 1].
10Common Mistakes to Avoid
Dividing by a Trig Function
Wrong: sin x cos x = cos x ⇒ sin x = 1 (divided by cos x)
Right: sin x cos x − cos x = 0 ⇒ cos x(sin x − 1) = 0. Solutions: cos x = 0 OR sin x = 1. Factor, don’t divide.
Using Wrong Period for General Solutions
Wrong: tan x = 1 ⇒ x = π/4 + 2πn
Right: tan x = 1 ⇒ x = π/4 + πn. Tangent has period π, not 2π. Using 2πn misses half the solutions.
Forgetting the ± When Taking Square Roots
Wrong: cos²x = 1/4 ⇒ cos x = 1/2
Right: cos x = ±1/2. Both positive and negative values are valid, giving solutions in all four quadrants.
Not Scaling the Interval for Multiple Angles
Wrong: Solving sin(2x) in [0, 2π) and only finding 2 solutions.
Right: For 2x, use the interval [0, 4π). This gives 4 solutions for x, not 2.
Including Extraneous Solutions
Wrong: Not checking if solutions make the original equation undefined.
Right: Always verify solutions in the original equation, especially when you squared both sides or when tangent/cotangent are involved.
Missing Quadrants
Wrong: sin x = −1/2 ⇒ x = 7π/6 only.
Right: sin is negative in QIII and QIV. x = 7π/6 and x = 11π/6. Always check all relevant quadrants.
11Quick Revision Summary
- Isolate the trig function first, then find the reference angle.
- Use ASTC to determine which quadrants contain solutions.
- General solutions: sin/cos use + 2πn, tan uses + πn.
- Factor instead of dividing by a trig function to avoid losing solutions.
- Use Pythagorean identity to convert between sin and cos.
- For quadratic forms, substitute u = trig(x) and solve the quadratic.
- For multiple angles (2x, 3x), scale the interval accordingly.
- Always include ± when taking square roots.
- If sin x or cos x equals a value outside [−1, 1], there is no solution.
- Always verify solutions in the original equation.
Frequently Asked Questions
Why add 2πn to solutions?
Trig functions are periodic. Adding 2πn (or πn for tangent) accounts for all coterminal angles.
How do I know radians vs. degrees?
Check the problem. If the interval is [0, 2π), use radians. If [0°, 360°), use degrees.
What if sin x = 1.5?
No solution — sine range is [−1, 1]. Tangent can be any real number.
When should I use identities?
When the equation has different trig functions (sin and cos) or different powers. The goal is a single trig function and a single power.
Practice Quiz
Test your knowledge — select the correct answer for each question.
1.What are the principal solutions for sin x = 1/2 in [0, 2π)?
2.Which is the general solution for cos x = 0?
3.Solve 2tan x − 2 = 0 for x in [0°, 360°).
4.First step to solve 2sin²x + cos x − 1 = 0?
5.If sin x = −1, the general solutions are:
6.Find all solutions for tan x · cos x = 0 in [0, 2π).
7.Which equation is quadratic in form?
8.Solving 2cos²x − 1 = 0, what are the values of cos x?
9.If 2x = π/3 + 2πn, then x = ?
10.What is a common mistake when solving trig equations?
Study Tips
- Draw the unit circle — visualizing where the angle falls makes selecting the correct quadrant much easier.
- Practice factoring trig expressions — treat sin x and cos x like algebraic variables.
- Always check your solutions — plug them back into the original equation to catch extraneous solutions.
- Know your identities cold — Pythagorean and double-angle identities are the most commonly needed.
- Write the general solution last — find principal solutions first, then add the period.