Trigonometric equations involve trigonometric functions such as sine, cosine, tangent, secant, cosecant, and cotangent. These equations involve finding the values of the trigonometric functions that satisfy the given equation.

Solving Trigonometric Equations

There are several methods to solve trigonometric equations, including:

Algebraic Manipulation: Rearranging the equation using algebraic techniques to isolate the trigonometric function.
Using Trigonometric Identities: Applying trigonometric identities to simplify the equation.
Graphical Methods: Graphing the trigonometric functions and finding the points of intersection with the given equation.
Using Special Angles: Utilizing the values of trigonometric functions for special angles (0°, 30°, 45°, 60°, 90°) to solve the equation.

Common Trigonometric Identities

It's important to be familiar with common trigonometric identities when solving trigonometric equations:

Pythagorean Identities: $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, $1 + \cot^2(\theta) = \csc^2(\theta)$
Sum and Difference Identities: $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$, $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$, etc.
Double Angle Identities: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$, etc.

Example Problem

Solve the trigonometric equation $\cos(x) = \frac{1}{2}$ for $0° \leq x \leq 360°$.

Solution:

We can use the special angles to find the solutions. The cosine function equals $\frac{1}{2}$ at $60°$ and $300°$. So, the solutions are $x = 60°$ and $x = 300°$.

It's important to check for any other possible solutions by adding or subtracting multiples of $360°$ to the initial solutions.