Trigonometric Equations Explained: Solving sin(x)=k, cos(x)=k, and tan(x)=k
PUC Mathematics | Trigonometric Equations | Daniel Classes
Why Solving Trig Equations Matters
Algebraic equations have solutions. Quadratic equations have 1 or 2. Polynomial equations have at most n solutions.
Trigonometric equations are different. Because sine and cosine repeat every 2π radians, a single equation like sin(x) = 0.5 has infinitely many solutions.
Core skill: Find ONE solution, then use periodicity to find all solutions.
Solving sin(x) = k
Method:
- If k ∈ [-1, 1], the equation has solutions. Otherwise, no solution.
- Find the reference angle: α = arcsin(|k|)
- Determine which quadrants have sin positive (quadrants I and II) or negative (III and IV)
- General solution: x = α + 2nπ or x = π – α + 2nπ (and variations depending on the sign of k)
Example: Solve sin(x) = 1/2
Reference angle: α = arcsin(1/2) = π/6
sin is positive in quadrants I and II, so:
x = π/6 + 2nπ or x = π – π/6 + 2nπ = 5π/6 + 2nπ
General solution: x = π/6 + 2nπ, 5π/6 + 2nπ (n ∈ ℤ)
Solving cos(x) = k
Method:
- If k ∈ [-1, 1], the equation has solutions. Otherwise, no solution.
- Find the reference angle: α = arccos(|k|)
- Determine which quadrants have cos positive (I and IV) or negative (II and III)
- General solution: x = ±α + 2nπ
Example: Solve cos(x) = 1/2
Reference angle: α = arccos(1/2) = π/3
cos is positive in quadrants I and IV, so:
x = π/3 + 2nπ or x = -π/3 + 2nπ = 2π – π/3 + 2nπ = 5π/3 + 2nπ
General solution: x = ±π/3 + 2nπ (n ∈ ℤ)
Solving tan(x) = k
Method:
- Find the reference angle: α = arctan(|k|)
- tan has period π, so solutions repeat every π (not 2π)
- General solution: x = α + nπ or x = -α + nπ (depending on sign)
Example: Solve tan(x) = 1
Reference angle: α = arctan(1) = π/4
General solution: x = π/4 + nπ (n ∈ ℤ)
Solving More Complex Equations
Example: Solve 2sin(x) + 1 = 0
Step 1: 2sin(x) = -1
Step 2: sin(x) = -1/2
Step 3: Reference angle: α = π/6
Step 4: sin is negative in quadrants III and IV
Step 5: x = π + π/6 + 2nπ = 7π/6 + 2nπ or x = 2π – π/6 + 2nπ = 11π/6 + 2nπ
General solution: x = 7π/6 + 2nπ, 11π/6 + 2nπ
How PUC Exams Test Trig Equations
Type 1: Basic Equations (2-3 marks)
Example: “Solve sin(x) = √3/2.”
Solution: x = π/3 + 2nπ, 2π/3 + 2nπ
Type 2: Transformed Equations (3-5 marks)
Example: “Solve 2sin(2x) = 1.”
Solution: sin(2x) = 1/2, so 2x = π/6 + 2nπ or 2x = 5π/6 + 2nπ, thus x = π/12 + nπ or x = 5π/12 + nπ
Type 3: Equations in a Specific Range (3-5 marks)
Example: “Solve sin(x) = 1/2 for x ∈ [0, 2π].”
Solution: x = π/6 or x = 5π/6 (just the solutions in the given range)
The Five Common Mistakes
Mistake 1: Forgetting the ±2nπ
Example: “Solve sin(x) = 1/2.”
Wrong: “x = π/6” [Missing all other solutions]
Right: “x = π/6 + 2nπ, 5π/6 + 2nπ”
Fix: Trig functions are periodic. Write the general solution with 2nπ (for sin/cos) or nπ (for tan).
Mistake 2: Confusing sin and cos Solutions
Example: “Solve cos(x) = 1/2.”
Wrong: Uses sin formula: x = π/3 + 2nπ, 2π/3 + 2nπ [Wrong]
Right: x = ±π/3 + 2nπ
Fix: sin solutions are π – α and α. cos solutions are ±α.
Mistake 3: Forgetting tan’s Period Is π, Not 2π
Example: “Solve tan(x) = 1.”
Wrong: “x = π/4 + 2nπ”
Right: “x = π/4 + nπ”
Fix: tan repeats every π, not 2π.
Mistake 4: Not Handling Transformations
Example: “Solve 2sin(x – π/4) = 1.”
Wrong: Solves sin(x) = 1/2 directly without accounting for the transformation.
Right: First let u = x – π/4. Solve sin(u) = 1/2. Then x = u + π/4.
Fix: Substitute to simplify, solve, then back-substitute.
Mistake 5: Forgetting the Constraint -1 ≤ sin(x) ≤ 1
Example: “Solve sin(x) = 2.”
Wrong: Tries to find a reference angle.
Right: “No solution (since sine is always between -1 and 1).”
Fix: Check the constraint before solving.
What to Do Next
To master basic equations: Solve sin(x) = k, cos(x) = k, tan(x) = k for 20 different values of k.
To master transformations: Solve 10 equations involving 2sin(3x + π/4) = 1 type problems.
To master ranges: Solve 10 equations with constraints like x ∈ [0, π] or x ∈ [0, 2π].
The Problem Vault Alternative
The Problem Vault has 80+ Trigonometric Equations problems.