Trigonometric Equations Solver

🎯 Master Trigonometric Equations!

Learn to solve different types of trigonometric equations step by step with interactive practice.

📚 What You'll Learn:

✅ Basic trigonometric equations
✅ Quadratic trigonometric equations
✅ Multiple angle equations
✅ Sum and difference equations
✅ Mixed trigonometric equations
✅ Step-by-step solving strategies

🧮 Interactive Features:

🎯 Step-by-step solutions
📝 Practice problems on each slide
✨ Instant feedback
🔄 Multiple solution methods
📊 Progress tracking
🎉 Interactive examples

🚀 Ready to become a trigonometric equation expert?

Each slide will teach you a new type of equation with hands-on practice!

📐 Type 1: Basic Sine Equations

General Form: \(\sin x = a\)

📋 Step-by-Step Method:

Step 1: Check if \(|a| \leq 1\)
Step 2: Find reference angle: \(\alpha = \arcsin|a|\)
Step 3: Determine quadrants where sine has the correct sign
Step 4: Write general solutions

Example: \(\sin x = \frac{1}{2}\)

Step 1: \(|\frac{1}{2}| = 0.5 \leq 1\) ✓

Step 2: \(\alpha = \arcsin(\frac{1}{2}) = 30°\)

Step 3: Sine is positive in Q1 and Q2

Step 4: \(x = 30° + 360°k\) or \(x = 150° + 360°k\)

🧮 Practice Problem

Solve: \(\sin x = \frac{\sqrt{3}}{2}\)

Enter the reference angle (in degrees):

Enter first solution (0° to 360°):

Enter second solution (0° to 360°):

📐 Type 2: Basic Cosine Equations

General Form: \(\cos x = a\)

📋 Step-by-Step Method:

Step 1: Check if \(|a| \leq 1\)
Step 2: Find reference angle: \(\alpha = \arccos|a|\)
Step 3: Determine quadrants where cosine has the correct sign
Step 4: Write general solutions

Example: \(\cos x = -\frac{1}{2}\)

Step 1: \(|-\frac{1}{2}| = 0.5 \leq 1\) ✓

Step 2: \(\alpha = \arccos(\frac{1}{2}) = 60°\)

Step 3: Cosine is negative in Q2 and Q3

Step 4: \(x = 120° + 360°k\) or \(x = 240° + 360°k\)

🧮 Practice Problem

Solve: \(\cos x = \frac{\sqrt{2}}{2}\)

Enter the reference angle (in degrees):

Enter first solution (0° to 360°):

Enter second solution (0° to 360°):

📐 Type 3: Basic Tangent Equations

General Form: \(\tan x = a\)

📋 Step-by-Step Method:

Step 1: Find reference angle: \(\alpha = \arctan|a|\)
Step 2: Determine quadrants where tangent has the correct sign
Step 3: Use period of 180° for tangent
Step 4: Write general solution

Example: \(\tan x = \sqrt{3}\)

Step 1: \(\alpha = \arctan(\sqrt{3}) = 60°\)

Step 2: Tangent is positive in Q1 and Q3

Step 3: Period = 180°

Step 4: \(x = 60° + 180°k\)

Solutions: \(x = 60°, 240°, ...\)

🧮 Practice Problem

Solve: \(\tan x = -1\)

Enter the reference angle (in degrees):

Enter first solution (0° to 360°):

Enter second solution (0° to 360°):

🔢 Type 4: Quadratic Trigonometric Equations

General Form: \(a\sin^2 x + b\sin x + c = 0\)

📋 Step-by-Step Method:

Step 1: Let \(u = \sin x\) (or \(\cos x\), \(\tan x\))
Step 2: Solve quadratic: \(au^2 + bu + c = 0\)
Step 3: Find values of \(u\)
Step 4: Solve basic trig equations for each \(u\)
Step 5: Check validity of solutions

Example: \(2\sin^2 x - \sin x - 1 = 0\)

Step 1: Let \(u = \sin x\)

Step 2: \(2u^2 - u - 1 = 0\)

Step 3: \((2u + 1)(u - 1) = 0\)

Step 4: \(u = -\frac{1}{2}\) or \(u = 1\)

Step 5: \(\sin x = -\frac{1}{2}\) or \(\sin x = 1\)

🧮 Practice Problem

Solve: \(\cos^2 x - \cos x = 0\)

Factor the equation. What are the two factors?

What are the values of cos x?

🔄 Type 5: Double Angle Equations

Forms: \(\sin 2x = a\), \(\cos 2x = a\), \(\tan 2x = a\)

📋 Step-by-Step Method:

Step 1: Solve for the double angle: \(2x = \text{solutions}\)
Step 2: Divide by 2 to get \(x\)
Step 3: Consider the period of the double angle
Step 4: List all solutions in the given interval

Example: \(\sin 2x = \frac{1}{2}\)

Step 1: \(2x = 30° + 360°k\) or \(2x = 150° + 360°k\)

Step 2: \(x = 15° + 180°k\) or \(x = 75° + 180°k\)

Step 3: For \([0°, 360°)\):

Step 4: \(x = 15°, 75°, 195°, 255°\)

🧮 Practice Problem

Solve: \(\cos 2x = -\frac{\sqrt{2}}{2}\) for \([0°, 360°)\)

First, solve for 2x. Enter the reference angle:

Enter the four solutions for x (separated by commas):

➗ Type 6: Half Angle Equations

Forms: \(\sin \frac{x}{2} = a\), \(\cos \frac{x}{2} = a\), \(\tan \frac{x}{2} = a\)

📋 Step-by-Step Method:

Step 1: Solve for the half angle: \(\frac{x}{2} = \text{solutions}\)
Step 2: Multiply by 2 to get \(x\)
Step 3: Consider the extended period
Step 4: List all solutions in the given interval

Example: \(\sin \frac{x}{2} = \frac{\sqrt{3}}{2}\)

Step 1: \(\frac{x}{2} = 60° + 360°k\) or \(\frac{x}{2} = 120° + 360°k\)

Step 2: \(x = 120° + 720°k\) or \(x = 240° + 720°k\)

Step 3: Period is now 720°

Step 4: For \([0°, 720°)\): \(x = 120°, 240°\)

🧮 Practice Problem

Solve: \(\cos \frac{x}{2} = \frac{1}{2}\) for \([0°, 720°)\)

What angle has cosine = 1/2?

Enter solutions for x/2 in [0°, 360°):

Enter final solutions for x:

➕ Type 7: Sum and Difference Equations

Forms: \(\sin(x + a) = b\), \(\cos(x - a) = b\)

📋 Step-by-Step Method:

Step 1: Solve for the compound angle: \((x + a) = \text{solutions}\)
Step 2: Isolate \(x\) by subtracting/adding \(a\)
Step 3: Apply the appropriate period
Step 4: List solutions in the given interval

Example: \(\sin(x + 30°) = \frac{1}{2}\)

Step 1: \(x + 30° = 30° + 360°k\) or \(x + 30° = 150° + 360°k\)

Step 2: \(x = 0° + 360°k\) or \(x = 120° + 360°k\)

Step 3: Period = 360°

Step 4: For \([0°, 360°)\): \(x = 0°, 120°\)

🧮 Practice Problem

Solve: \(\cos(x - 45°) = \frac{\sqrt{2}}{2}\) for \([0°, 360°)\)

What angle has cosine = √2/2?

Solve for (x - 45°). Enter the two values:

Enter final solutions for x:

✖️ Type 8: Product-to-Sum Equations

Forms: \(\sin A \cos B = c\), \(\cos A \sin B = c\)

📋 Product-to-Sum Formulas:

\(\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]\)

\(\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]\)

\(\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]\)

\(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\)

Example: \(\sin 3x \cos x = \frac{1}{2}\)

Step 1: Use formula: \(\sin 3x \cos x = \frac{1}{2}[\sin 4x + \sin 2x]\)

Step 2: \(\frac{1}{2}[\sin 4x + \sin 2x] = \frac{1}{2}\)

Step 3: \(\sin 4x + \sin 2x = 1\)

Step 4: Solve the resulting equation

🧮 Practice Problem

Convert to sum form: \(\cos 5x \cos 2x\)

What is A + B?

What is A - B?

Write the complete sum form:

🔀 Type 9: Mixed Trigonometric Equations

Forms: Equations with multiple trig functions

📋 Strategy Selection:

Strategy 1: Use Pythagorean identities
Strategy 2: Factor common terms
Strategy 3: Convert to single function
Strategy 4: Use substitution

Example: \(\sin x + \cos x = 1\)

Method: Square both sides

Step 1: \((\sin x + \cos x)^2 = 1\)

Step 2: \(\sin^2 x + 2\sin x \cos x + \cos^2 x = 1\)

Step 3: \(1 + 2\sin x \cos x = 1\)

Step 4: \(\sin x \cos x = 0\)

Step 5: \(\sin x = 0\) or \(\cos x = 0\)

🧮 Practice Problem

Solve: \(\sin^2 x + \cos^2 x = \sin x\)

Use the Pythagorean identity. What does sin²x + cos²x equal?

Simplify the equation. What does sin x equal?

Enter the solutions for x in [0°, 360°):

🎯 Advanced Solving Techniques

🔧 Key Strategies Summary

1. Substitution Method

Use \(u = \sin x\), \(u = \cos x\), or \(u = \tan x\) for quadratic forms

2. Factoring

Factor out common trigonometric functions

3. Identities

Use Pythagorean, double angle, and sum formulas

4. Graphical Analysis

Visualize intersections and periods

🧮 Challenge Problem

Solve: \(2\sin^2 x - 3\sin x + 1 = 0\)

Let u = sin x. Write the quadratic equation:

Factor the quadratic:

Values of u (sin x):

Final solutions for x in [0°, 360°):

🏆 Final Challenge Quiz

Question 1: Solve \(\sin x = -\frac{\sqrt{3}}{2}\) for \([0°, 360°)\)

Question 2: Solve \(\tan^2 x - 1 = 0\) for \([0°, 360°)\)

Question 3: Solve \(\sin 2x = 1\) for \([0°, 360°)\)

🎯 Master Trigonometric Equations!

📚 What You'll Learn:

🧮 Interactive Features:

📐 Type 1: Basic Sine Equations

General Form: \(\sin x = a\)

📋 Step-by-Step Method:

Example: \(\sin x = \frac{1}{2}\)

🧮 Practice Problem

📐 Type 2: Basic Cosine Equations

General Form: \(\cos x = a\)

📋 Step-by-Step Method:

Example: \(\cos x = -\frac{1}{2}\)

🧮 Practice Problem

📐 Type 3: Basic Tangent Equations

General Form: \(\tan x = a\)

📋 Step-by-Step Method:

Example: \(\tan x = \sqrt{3}\)

🧮 Practice Problem

🔢 Type 4: Quadratic Trigonometric Equations

General Form: \(a\sin^2 x + b\sin x + c = 0\)

📋 Step-by-Step Method:

Example: \(2\sin^2 x - \sin x - 1 = 0\)

🧮 Practice Problem

🔄 Type 5: Double Angle Equations

Forms: \(\sin 2x = a\), \(\cos 2x = a\), \(\tan 2x = a\)

📋 Step-by-Step Method:

Example: \(\sin 2x = \frac{1}{2}\)

🧮 Practice Problem

➗ Type 6: Half Angle Equations

Forms: \(\sin \frac{x}{2} = a\), \(\cos \frac{x}{2} = a\), \(\tan \frac{x}{2} = a\)

📋 Step-by-Step Method:

Example: \(\sin \frac{x}{2} = \frac{\sqrt{3}}{2}\)

🧮 Practice Problem

➕ Type 7: Sum and Difference Equations

Forms: \(\sin(x + a) = b\), \(\cos(x - a) = b\)

📋 Step-by-Step Method:

Example: \(\sin(x + 30°) = \frac{1}{2}\)

🧮 Practice Problem

✖️ Type 8: Product-to-Sum Equations

Forms: \(\sin A \cos B = c\), \(\cos A \sin B = c\)

📋 Product-to-Sum Formulas:

Example: \(\sin 3x \cos x = \frac{1}{2}\)

🧮 Practice Problem

🔀 Type 9: Mixed Trigonometric Equations

Forms: Equations with multiple trig functions

📋 Strategy Selection:

Example: \(\sin x + \cos x = 1\)

🧮 Practice Problem

🎯 Advanced Solving Techniques

🔧 Key Strategies Summary

1. Substitution Method

2. Factoring

3. Identities

4. Graphical Analysis

🧮 Challenge Problem

🏆 Final Challenge Quiz

🎉 Quiz Complete!