General Strategy
Isolate
Get the trig function alone on one side
Reference angle
Find arcsin/arccos/arctan of the absolute value
Quadrants
Determine which quadrants match the sign of the value
List solutions
Write all solutions in [0, 2π) using the quadrant angles
General form
Add +2πk (or +πk for tan) for all solutions
| Function | Positive in quadrants | Negative in quadrants | Period |
|---|---|---|---|
| sin(x) | Q1, Q2 (y > 0) | Q3, Q4 (y < 0) | 2π |
| cos(x) | Q1, Q4 (x > 0) | Q2, Q3 (x < 0) | 2π |
| tan(x) | Q1, Q3 | Q2, Q4 | π |
Worked Examples
Example 1 — Basic Sine
Solve sin(x) = √3/2 on [0, 2π)
Step 1: sin is already isolated
Step 2: Reference angle = arcsin(√3/2) = π/3
Step 3: sin(√3/2) > 0 → Q1 and Q2
Step 4:
Q1: x = π/3
Q2: x = π − π/3 = 2π/3
Solutions: x = π/3, 2π/3
General: x = π/3 + 2πk or x = 2π/3 + 2πk
Example 2 — Negative Value
Solve cos(x) = −1/2 on [0, 2π)
Reference angle: arccos(1/2) = π/3
Sign: cos < 0 → Q2 and Q3
Q2: x = π − π/3 = 2π/3
Q3: x = π + π/3 = 4π/3
Solutions: x = 2π/3, 4π/3
Example 3 — Requires Algebra First
Solve 2sin(x) + 1 = 0 on [0, 2π)
2sin(x) = −1 → sin(x) = −1/2
Reference angle: arcsin(1/2) = π/6
Sign: sin < 0 → Q3 and Q4
Q3: x = π + π/6 = 7π/6
Q4: x = 2π − π/6 = 11π/6
Solutions: x = 7π/6, 11π/6
Example 4 — Tan Equation
Solve tan(x) = −√3 on [0, 2π)
Reference angle: arctan(√3) = π/3
Sign: tan < 0 → Q2 and Q4
Q2: x = π − π/3 = 2π/3
Q4: x = 2π − π/3 = 5π/3
Solutions: x = 2π/3, 5π/3
General: x = 2π/3 + πk (tan period is π, so one expression covers both)
Quadratic Trig Equations
When a trig function is squared, treat it as a quadratic in u = sin(x), u = cos(x), or u = tan(x).
Example 5 — Quadratic in sin
Solve 2sin²(x) − sin(x) − 1 = 0 on [0, 2π)
Let u = sin(x): 2u² − u − 1 = 0
Factor: (2u + 1)(u − 1) = 0
u = −1/2 or u = 1
Case 1: sin(x) = −1/2
x = 7π/6 or x = 11π/6
Case 2: sin(x) = 1
x = π/2
Solutions: x = π/2, 7π/6, 11π/6
Example 6 — Using Pythagorean Identity
Solve cos²(x) − sin(x) − 1 = 0 on [0, 2π)
Replace cos²(x) using identity: cos²(x) = 1 − sin²(x)
1 − sin²(x) − sin(x) − 1 = 0
−sin²(x) − sin(x) = 0
sin(x)(−sin(x) − 1) = 0
sin(x) = 0 or sin(x) = −1
sin(x) = 0: x = 0, π
sin(x) = −1: x = 3π/2
Solutions: x = 0, π, 3π/2
General Solution Format
| Equation | Solutions on [0,2π) | General Solution |
|---|---|---|
| sin(x) = c, 0 < c < 1 | α, π−α | α + 2πk or (π−α) + 2πk |
| sin(x) = c, −1 < c < 0 | π+α, 2π−α (α = arcsin|c|) | (π+α) + 2πk or (2π−α) + 2πk |
| cos(x) = c, 0 < c < 1 | α, 2π−α | α + 2πk or (2π−α) + 2πk |
| cos(x) = c, −1 < c < 0 | π−α, π+α | (π−α) + 2πk or (π+α) + 2πk |
| tan(x) = c | α, π+α (in [0,2π)) | α + πk |
| sin(x) = 1 | π/2 | π/2 + 2πk |
| sin(x) = −1 | 3π/2 | 3π/2 + 2πk |
| cos(x) = 1 | 0 | 2πk |
| cos(x) = −1 | π | π + 2πk |
Exam Strategy
Check how many solutions are expected
If the problem says [0, 2π), find 1–2 solutions per isolated equation. If it says 'all solutions,' add 2πk (or πk for tan). Missing the general form costs full credit.
Don't divide both sides by a trig function
Dividing sin(x)cos(x) = 0 by cos(x) loses the solutions where cos(x) = 0. Always factor instead of dividing — move everything to one side and factor out.
Use identities to get one function
If you see sin and cos mixed, use sin²x + cos²x = 1 to convert to one function. Then it becomes a quadratic you can factor or use the quadratic formula on.
Frequently Asked Questions
What is the general solution of a trig equation?
The general solution gives ALL values of x that satisfy the equation, not just those in [0,2π). Since trig functions are periodic, you add integer multiples of the period. For sin and cos (period 2π), write x = solution + 2πk where k is any integer. For tan (period π), write x = solution + πk.
How many solutions does sin(x) = c have on [0, 2π)?
If |c| < 1, sin(x) = c has exactly 2 solutions on [0,2π). The reference angle is arcsin(|c|). The two solutions are in the quadrants where sin has the appropriate sign: for c > 0, Q1 and Q2; for c < 0, Q3 and Q4. If c = 1, there's 1 solution (π/2). If c = -1, there's 1 solution (3π/2). If |c| > 1, there are no solutions.
When do I need to use a trig identity to solve an equation?
Use a trig identity when the equation has more than one trig function (e.g., sin and cos together) or when it's a quadratic in a trig function that doesn't factor easily. Common approaches: use sin²x + cos²x = 1 to convert everything to one function, use double-angle formulas to reduce degree, or factor after substituting u = sin(x) or u = cos(x).
Related Topics
Practice Trig Equation Problems
Work through trig equations with step-by-step solutions and AI explanations — free on NailTheTest.
Start Practicing Free