When solving trigonometric equations, we use several key techniques:

1. Isolate the trigonometric function
2. Use inverse trig functions or the unit circle
3. Find all solutions in the given interval
4. Consider periodicity for general solutions

Types of Trigonometric Equations

Type 1: Linear Equations

Example form: $\sin(x) = a$ where $-1 \leq a \leq 1$

Solution method:

1. Find the reference angle: $\theta_r = \arcsin(|a|)$
2. Determine which quadrants based on the sign of $a$
3. Find all solutions in $[0, 2\pi)$
4. Add $2\pi n$ for general solution

General solution patterns:

• If $\sin(x) = a$: $x = \arcsin(a) + 2\pi n$ or $x = \pi - \arcsin(a) + 2\pi n$
• If $\cos(x) = a$: $x = \arccos(a) + 2\pi n$ or $x = -\arccos(a) + 2\pi n$
• If $\tan(x) = a$: $x = \arctan(a) + \pi n$

Type 2: Quadratic Equations

Example form: $\sin^2(x) + b\sin(x) + c = 0$

Solution method:

1. Substitute $u = \sin(x)$ (or the relevant trig function)
2. Solve the quadratic equation for $u$
3. Solve $\sin(x) = u$ for each value of $u$
4. Check that solutions are in the domain

Type 3: Equations with Multiple Angles

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

Solution method:

1. Let $u = 2x$ (or the multiple angle)
2. Solve $\sin(u) = \frac{1}{2}$
3. Divide by the coefficient to find $x$
4. Consider the extended period

Type 4: Equations with Multiple Functions

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

Solution methods:

• Use Pythagorean identities to express in terms of one function
• Square both sides (check for extraneous solutions)
• Use sum-to-product or product-to-sum identities

Key Identities for Solving

Pythagorean Identities

• $\sin^2(x) + \cos^2(x) = 1$
• $1 + \tan^2(x) = \sec^2(x)$
• $1 + \cot^2(x) = \csc^2(x)$

Double Angle Formulas

• $\sin(2x) = 2\sin(x)\cos(x)$
• $\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)$
• $\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$

Half Angle Formulas

• $\sin^2(x) = \frac{1 - \cos(2x)}{2}$
• $\cos^2(x) = \frac{1 + \cos(2x)}{2}$

Interval Considerations

• Standard interval: $[0, 2\pi)$ or $[0°, 360°)$
• Extended intervals: May need to consider multiple periods
• General solution: Include $+ 2\pi n$ (or $+ \pi n$ for tangent) where $n$ is an integer

Checking Solutions

Always verify solutions by:

1. Substituting back into the original equation
2. Checking domain restrictions (e.g., no division by zero)
3. Eliminating extraneous solutions from squaring

Common Reference Angles

Step 2: Find solutions using the unit circle

$\sin(x) = \frac{1}{2}$ when $x$ is in Quadrants I and II (where sine is positive).

Reference angle: $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$

Step 3: Find all solutions in $[0, 2\pi)$

• Quadrant I: $x = \frac{\pi}{6}$
• Quadrant II: $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$

Answer: $x = \frac{\pi}{6}, \frac{5\pi}{6}$

Verification:

• $2\sin(\frac{\pi}{6}) - 1 = 2(\frac{1}{2}) - 1 = 0$ ✓
• $2\sin(\frac{5\pi}{6}) - 1 = 2(\frac{1}{2}) - 1 = 0$ ✓