Trigonometric Equations & General Solutions

TAKEAWAYS:

• Solve basic trigonometric equations (e.g., sin θ = 1/2) by finding principal values and adding periodicity.
• Use identities (e.g., sin²θ + cos²θ = 1) to rewrite equations before solving.
• General solutions for sin, cos, tan follow patterns: θ = nπ + α, θ = 2nπ ± α, or θ = nπ + α.
• Periodicity (repetition every 2π or π) helps find all possible solutions.
• Quadrant analysis determines signs of trigonometric functions.
• NEB exam focus: Solve equations, find general solutions, and verify answers.

---

1. Basic Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sin θ, cos θ, tan θ. The goal is to find the value(s) of θ that satisfy the equation.

Key Concepts

1. Principal Values: The smallest angle (in the range [0, 2π) or [-π/2, π/2]) that satisfies the equation.
2. General Solutions: All possible solutions, considering the periodic nature of trigonometric functions.
3. Periodicity:
   • sin θ and cos θ repeat every 2π (360°).
   • tan θ repeats every π (180°).

Example 1: Solve sin θ = 1/2

Step 1: Find the principal value(s) of θ where sin θ = 1/2. From the unit circle, sin θ = 1/2 at:

• θ = π/6 (30°) in the first quadrant.
• θ = 5π/6 (150°) in the second quadrant (since sin is positive in Q1 and Q2).

Step 2: Write the general solution. Since sin θ has a period of 2π, all solutions are: θ = π/6 + 2nπ or θ = 5π/6 + 2nπ, where n ∈ ℤ (n is any integer).

Visualization:

Example 2: Solve cos θ = -√3/2

Step 1: Find principal values. cos θ = -√3/2 at:

• θ = 5π/6 (150°) in the second quadrant.
• θ = 7π/6 (210°) in the third quadrant (cos is negative in Q2 and Q3).

Step 2: General solution. cos θ has a period of 2π, so: θ = 5π/6 + 2nπ or θ = 7π/6 + 2nπ, where n ∈ ℤ.

---

2. Solving Using Identities

Some equations require rewriting using trigonometric identities before solving.

Example 3: Solve sin²θ = 3/4

Step 1: Rewrite using the identity sin²θ + cos²θ = 1. sin²θ = 3/4 ⇒ cos²θ = 1 - 3/4 = 1/4 ⇒ cos θ = ±1/2.

Step 2: Solve cos θ = 1/2 and cos θ = -1/2 separately.

• For cos θ = 1/2: θ = π/3 + 2nπ or θ = 5π/3 + 2nπ.
• For cos θ = -1/2: θ = 2π/3 + 2nπ or θ = 4π/3 + 2nπ.

Final Solution: θ = π/3 + nπ or θ = 2π/3 + nπ (combining all solutions).

Why? Notice that the solutions repeat every π (not 2π) because squaring removes the sign difference.

---

3. General Solutions for tan θ

The tangent function has a period of π, so its general solution is simpler.

Example 4: Solve tan θ = √3

Step 1: Find the principal value. tan θ = √3 ⇒ θ = π/3 (60°).

Step 2: General solution. Since tan θ repeats every π: θ = π/3 + nπ, where n ∈ ℤ.

Visualization:

---

4. Quadrant Analysis

The sign of trigonometric functions depends on the quadrant:

Example 5: Solve sin θ = -1/2 Step 1: Principal values where sin θ = -1/2.

• θ = 7π/6 (210°) in Q3.
• θ = 11π/6 (330°) in Q4.

Step 2: General solution. θ = 7π/6 + 2nπ or θ = 11π/6 + 2nπ, where n ∈ ℤ.

---

5. Solving Using Multiple Angles

Some equations involve multiple angles (e.g., sin 2θ, cos 3θ). Use substitution to simplify.

Example 6: Solve sin 2θ = √2/2

Step 1: Let φ = 2θ ⇒ sin φ = √2/2. Principal solutions for φ: φ = π/4 + 2nπ or φ = 3π/4 + 2nπ.

Step 2: Substitute back θ = φ/2. θ = π/8 + nπ or θ = 3π/8 + nπ.

Verification: Check θ = π/8 in sin 2θ: sin(2 × π/8) = sin(π/4) = √2/2 ✔️.

---

6. Comparing sin, cos, and tan Equations

Example 7: Solve cos 2θ = 0 Step 1: Principal solutions for cos φ = 0 (let φ = 2θ). φ = π/2 + nπ ⇒ 2θ = π/2 + nπ ⇒ θ = π/4 + nπ/2.

Final Answer: θ = π/4 + nπ/2, where n ∈ ℤ.

---

7. Applications of Trigonometric Equations

1. Engineering: Designing waves, oscillations, and signals.
2. Astronomy: Calculating planetary positions.
3. Physics: Analyzing harmonic motion.
4. Navigation: Determining directions using bearings.

---

Exam Tip

1. Always find principal values first before generalizing.
2. Check the range (e.g., [0, 2π]) if specified.
3. Use identities to simplify equations (e.g., sin²θ = 1 - cos²θ).
4. Verify solutions by plugging back into the original equation.
5. Memorize key angles:
   • sin(π/6) = 1/2, cos(π/3) = 1/2, tan(π/4) = 1.
6. NEB often tests:
   • Solving sin θ = a, cos θ = b, tan θ = c.
   • General solutions for all three functions.
   • Equations involving identities (e.g., sin²θ + cos²θ = 1).

---

NEB Board-Style Questions

Short Answer (5 marks each)

1. Solve the following equations for θ ∈ [0, 2π]: a) sin θ = -√3/2 b) cos 2θ = 1/2 c) tan(θ/2) = √3

2. Find the general solution for: a) sin θ = 1/√2 b) cos θ = -1 c) tan θ = -1

3. Solve: sin²θ - cos²θ = 0.

Long Answer (10 marks)

4. Solve the equation: 2 sin²θ + 3 cos θ = 0 for θ ∈ [0, 2π]. Verify your solutions.

5. Find all θ ∈ [0, 2π] such that: a) sin θ = cos θ b) tan θ = sin θ

---

Answer Key (Short Questions) 1a) θ = 4π/3, 5π/3 1b) θ = π/6, 5π/6, 7π/6, 11π/6 1c) θ = π/3, 5π/3

2a) θ = π/4 + 2nπ or 3π/4 + 2nπ 2b) θ = π + 2nπ 2c) θ = -π/4 + nπ

3. θ = π/4 + nπ/2