Solving Trigonometric Equations

Basic Trigonometric Equations

Solve for angle θ when given trig value

Use inverse trig functions:

• sin(θ) = a → θ = sin⁻¹(a)
• cos(θ) = a → θ = cos⁻¹(a)
• tan(θ) = a → θ = tan⁻¹(a)

Important: Find ALL solutions in given interval

Example 1: Simple Sine Equation

Solve: sin(θ) = 1/2 for 0 ≤ θ < 2π

Reference angle: sin⁻¹(1/2) = π/6

Sine positive in Quadrants I and II:

• Q1: θ = π/6
• Q2: θ = π - π/6 = 5π/6

Answer: θ = π/6, 5π/6

Example 2: Cosine Equation

Solve: cos(θ) = -1/2 for 0 ≤ θ < 2π

Reference angle: cos⁻¹(1/2) = π/3

Cosine negative in Quadrants II and III:

• Q2: θ = π - π/3 = 2π/3
• Q3: θ = π + π/3 = 4π/3

Answer: θ = 2π/3, 4π/3

Example 3: Tangent Equation

Solve: tan(θ) = √3 for 0 ≤ θ < 2π

Reference angle: tan⁻¹(√3) = π/3

Tangent positive in Quadrants I and III:

• Q1: θ = π/3
• Q3: θ = π + π/3 = 4π/3

Answer: θ = π/3, 4π/3

Using Algebra First

Isolate trig function before using inverse

Example 1: Addition

Solve: 2sin(θ) + 1 = 0 for 0 ≤ θ < 2π

Isolate sin(θ):

2sin(θ) = -1
sin(θ) = -1/2

Reference angle: π/6

Sine negative in Q3 and Q4:

• Q3: θ = π + π/6 = 7π/6
• Q4: θ = 2π - π/6 = 11π/6

Answer: θ = 7π/6, 11π/6

Example 2: Multiplication and Square Root

Solve: 3cos²(θ) = 1 for 0 ≤ θ < 2π

Solve for cos(θ):

cos²(θ) = 1/3
cos(θ) = ±√(1/3) = ±√3/3

For cos(θ) = √3/3 (positive in Q1, Q4):

• θ = cos⁻¹(√3/3) ≈ 0.955, 2π - 0.955 ≈ 5.328

For cos(θ) = -√3/3 (negative in Q2, Q3):

• θ = π - 0.955 ≈ 2.186, π + 0.955 ≈ 4.097

Answer: Four solutions in [0, 2π)

Quadratic Form Equations

Equations like a·sin²(θ) + b·sin(θ) + c = 0

Treat like quadratic: Let u = sin(θ), solve for u, then find θ

Example 1: Factor

Solve: 2sin²(θ) - sin(θ) = 0 for 0 ≤ θ < 2π

Factor:

sin(θ)(2sin(θ) - 1) = 0

Two cases:

Case 1: sin(θ) = 0

• θ = 0, π

Case 2: 2sin(θ) - 1 = 0 → sin(θ) = 1/2

• θ = π/6, 5π/6

Answer: θ = 0, π/6, 5π/6, π

Example 2: Quadratic Formula

Solve: 2cos²(θ) + 3cos(θ) - 2 = 0

Let u = cos(θ):

2u² + 3u - 2 = 0

Factor: (2u - 1)(u + 2) = 0

Solutions:

• u = 1/2 → cos(θ) = 1/2 → θ = π/3, 5π/3
• u = -2 → cos(θ) = -2 (impossible, |cos| ≤ 1)

Answer: θ = π/3, 5π/3

Using Identities to Solve

Rewrite equation using identities

Example 1: Pythagorean Identity

Solve: sin²(θ) + 2cos(θ) - 2 = 0 for 0 ≤ θ < 2π

Replace sin²(θ):

(1 - cos²(θ)) + 2cos(θ) - 2 = 0
-cos²(θ) + 2cos(θ) - 1 = 0
cos²(θ) - 2cos(θ) + 1 = 0

Factor:

(cos(θ) - 1)² = 0
cos(θ) = 1

Solve:

θ = 0

Answer: θ = 0

Example 2: Double-Angle Identity

Solve: cos(2θ) = cos(θ) for 0 ≤ θ < 2π

Expand cos(2θ):

2cos²(θ) - 1 = cos(θ)
2cos²(θ) - cos(θ) - 1 = 0

Factor:

(2cos(θ) + 1)(cos(θ) - 1) = 0

Case 1: 2cos(θ) + 1 = 0 → cos(θ) = -1/2

• θ = 2π/3, 4π/3

Case 2: cos(θ) - 1 = 0 → cos(θ) = 1

• θ = 0

Answer: θ = 0, 2π/3, 4π/3

Equations with Multiple Angles

Solve for the argument first, then find θ

Example 1: Double Angle

Solve: sin(2θ) = √3/2 for 0 ≤ θ < 2π

Let u = 2θ, solve for u in [0, 4π):

sin(u) = √3/2
u = π/3, 2π/3, 7π/3, 8π/3

Find θ = u/2:

θ = π/6, π/3, 7π/6, 4π/3

Answer: θ = π/6, π/3, 7π/6, 4π/3

Example 2: Different Coefficient

Solve: cos(3θ) = 1/2 for 0 ≤ θ < 2π

Let u = 3θ, solve in [0, 6π):

cos(u) = 1/2
u = π/3, 5π/3, 7π/3, 11π/3, 13π/3, 17π/3

Find θ = u/3:

θ = π/9, 5π/9, 7π/9, 11π/9, 13π/9, 17π/9

Answer: Six solutions

Equations with Different Functions

Convert to single function using identities

Example: Sine and Cosine

Solve: sin(θ) = cos(θ) for 0 ≤ θ < 2π

Divide both sides by cos(θ):

sin(θ)/cos(θ) = 1
tan(θ) = 1

Solve:

θ = π/4, 5π/4

Answer: θ = π/4, 5π/4

Finding All Solutions

General solutions include all rotations

Pattern: θ = θ₀ + 2πn (for sine/cosine)

Pattern: θ = θ₀ + πn (for tangent, shorter period)

Where n is any integer

Example: General Solution

Solve: sin(θ) = 1/2 (all solutions)

Principal solutions: θ = π/6, 5π/6

General form:

θ = π/6 + 2πn  or  θ = 5π/6 + 2πn

where n is any integer

Degree vs Radian Mode

Always check if answer should be in degrees or radians!

Example: Degrees

Solve: cos(θ) = √2/2 for 0° ≤ θ < 360°

Reference angle: 45°

Cosine positive in Q1, Q4:

θ = 45°, 315°

No Solution Cases

Sometimes equations have no solution

Example 1: Out of Range

Solve: sin(θ) = 2

Range of sine is [-1, 1]

Answer: No solution

Example 2: After Simplification

Solve: 2cos²(θ) + 5 = 0

Isolate:

cos²(θ) = -5/2

Cannot be negative

Answer: No solution

Real-World Applications

Physics: Find times when oscillating object at specific position

Engineering: Solve for phase angles in AC circuits

Astronomy: Determine positions of celestial objects

Signal processing: Find frequencies

Example: Ferris Wheel

Height on Ferris wheel: h(t) = 20sin(πt/30) + 25 meters

When is height 35 m?

Solve:

35 = 20sin(πt/30) + 25
10 = 20sin(πt/30)
sin(πt/30) = 1/2
πt/30 = π/6 + 2πn  or  5π/6 + 2πn
t = 5 + 60n  or  t = 25 + 60n

First times: t = 5 seconds and t = 25 seconds

Practice