Trigonometric Equations: Solution of Trigonometric Equations in a Given Interval

Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, tangent, etc. Solving these equations often requires finding the angles (or radians) that satisfy the equation within a specific interval. The solutions to trigonometric equations are not always unique, and they can repeat periodically due to the cyclic nature of trigonometric functions.

Basic Trigonometric Functions and Their Properties

Before we delve into solving trigonometric equations, let's review the basic trigonometric functions and their properties:

• Sine function: $\sin(\theta)$
• Cosine function: $\cos(\theta)$
• Tangent function: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Each of these functions has a period after which they repeat their values:

• The sine and cosine functions have a period of $2\pi$ radians (or $360^\circ$).
• The tangent function has a period of $\pi$ radians (or $180^\circ$).

General Solutions of Trigonometric Equations

The general solutions for basic trigonometric equations can be expressed as follows:

Function	Equation	General Solution
Sine	$\sin(\theta) = a$	$\theta = \sin^{-1}(a) + 2n\pi$ or $\theta = \pi - \sin^{-1}(a) + 2n\pi$, where $n$ is an integer
Cosine	$\cos(\theta) = a$	$\theta = \cos^{-1}(a) + 2n\pi$ or $\theta = -\cos^{-1}(a) + 2n\pi$, where $n$ is an integer
Tangent	$\tan(\theta) = a$	$\theta = \tan^{-1}(a) + n\pi$, where $n$ is an integer

Solving Trigonometric Equations in a Given Interval

When solving trigonometric equations, we are often interested in finding solutions within a specific interval, such as $[0, 2\pi]$ or $[-\pi, \pi]$. To find solutions within a given interval, we use the general solutions and then determine which of those solutions fall within the desired interval.

Steps to Solve Trigonometric Equations in a Given Interval

1. Isolate the Trigonometric Function: Try to express the equation with the trigonometric function isolated on one side.

2. Find the General Solution: Use the general solutions provided in the table above to find the general solution of the equation.

3. Determine Specific Solutions: Plug in different integer values for $n$ to find specific solutions that fall within the given interval.

4. Check for Extraneous Solutions: Verify that the solutions obtained are valid by plugging them back into the original equation.

Examples

Example 1: Solve $\sin(x) = \frac{\sqrt{2}}{2}$ in the interval $[0, 2\pi]$.

Step 1: The trigonometric function is already isolated.

Step 2: The general solution is $x = \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) + 2n\pi$ or $x = \pi - \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) + 2n\pi$.

Step 3: We know that $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$. So, the specific solutions are:

$x = \frac{\pi}{4} + 2n\pi$ or $x = \frac{3\pi}{4} + 2n\pi$.

For $n = 0$, we get $x = \frac{\pi}{4}$ and $x = \frac{3\pi}{4}$, both of which are in the interval $[0, 2\pi]$.

For $n = 1$, we get $x = \frac{\pi}{4} + 2\pi$ and $x = \frac{3\pi}{4} + 2\pi$, which are not in the interval $[0, 2\pi]$.

Step 4: No extraneous solutions in this case.

Example 2: Solve $\cos(x) = -1$ in the interval $[0, 2\pi]$.

Step 1: The trigonometric function is already isolated.

Step 2: The general solution is $x = \cos^{-1}(-1) + 2n\pi$ or $x = -\cos^{-1}(-1) + 2n\pi$.

Step 3: We know that $\cos^{-1}(-1) = \pi$. So, the specific solution is:

$x = \pi + 2n\pi$.

For $n = 0$, we get $x = \pi$, which is in the interval $[0, 2\pi]$.

For $n = 1$, we get $x = \pi + 2\pi$, which is not in the interval $[0, 2\pi]$.

Step 4: No extraneous solutions in this case.

Tips for Solving Trigonometric Equations

• Always start by simplifying the equation as much as possible.
• Use trigonometric identities to transform complex equations into simpler ones.
• Remember that the inverse trigonometric functions have limited ranges, so make sure to consider all possible angles that give the same sine, cosine, or tangent value.
• Be careful with the domain restrictions of the trigonometric functions, especially when dealing with the tangent function, which is undefined for certain values of $\theta$.

By following these steps and tips, you can effectively solve trigonometric equations within a given interval and find all possible solutions that satisfy the original equation.