Solving Trigonometric Equations: Example Problem

Solving Trigonometric Equations

Example Problem: Solving for x in a Trigonometric Equation

This section demonstrates how to solve a basic trigonometric equation, a key skill in Precalculus when working with trigonometric identities and equations.

• Key Point 1: To solve a trigonometric equation, isolate the trigonometric function and use inverse operations to find the solution for the variable.

• Key Point 2: Consider all possible solutions within the specified interval, as trigonometric functions are periodic.

Example:

Solve the equation $8 \sin x = 2\cos x$ for $x$.

• Step 1: Isolate $\sin x$:

$8 \sin x = 2 \cos x$

$\sin x = \frac{2}{8} \cos x$

$\sin x = \frac{1}{4} \cos x$

• Step 2: Divide both sides by $\cos x$ (assuming $\cos x \neq 0$):

$\frac{\sin x}{\cos x} = \frac{1}{4}$

$\tan x = \frac{1}{4}$

• Step 3: Solve for $x$:

$x = \arctan\left(\frac{1}{4}\right) + k\pi$, where $k$ is any integer (since tangent has a period of $\pi$).

• Step 4: If a specific interval is given (e.g., $0 \leq x < 2\pi$), list all solutions in that interval.

Additional info: This example illustrates the process of converting a trigonometric equation to a basic form and using inverse trigonometric functions to find solutions. Always check for extraneous solutions, especially when dividing by trigonometric functions.