Trigonometric Equations: Solution of Trigonometric Equations - General Solution

Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, tangent, etc. Solving these equations often requires finding all the angles that satisfy the equation, which is known as the general solution.

Basic Trigonometric Functions

Before diving into the general solutions, 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. For sine and cosine, the period is $2\pi$, and for tangent, the period is $\pi$.

General Solution of Trigonometric Equations

The general solution of a trigonometric equation takes into account all possible solutions within the domain of the trigonometric function. For example, the general solution for a sine equation $\sin(\theta) = a$ is given by:

• If $-1 \leq a \leq 1$, $\theta = \sin^{-1}(a) + 2n\pi$ or $\theta = \pi - \sin^{-1}(a) + 2n\pi$, where $n$ is an integer.
• If $a < -1$ or $a > 1$, there is no solution since the range of sine function is $[-1, 1]$.

Similarly, for cosine and tangent functions, we have:

• Cosine: $\cos(\theta) = a$ has solutions $\theta = \cos^{-1}(a) + 2n\pi$ or $\theta = -\cos^{-1}(a) + 2n\pi$.
• Tangent: $\tan(\theta) = a$ has solutions $\theta = \tan^{-1}(a) + n\pi$.

Table of General Solutions

Function	Equation	General Solution	Range of Principal Value
Sine	$\sin(\theta) = a$	$\theta = \sin^{-1}(a) + 2n\pi$ or $\theta = \pi - \sin^{-1}(a) + 2n\pi$	$[-\frac{\pi}{2}, \frac{\pi}{2}]$
Cosine	$\cos(\theta) = a$	$\theta = \cos^{-1}(a) + 2n\pi$ or $\theta = -\cos^{-1}(a) + 2n\pi$	$[0, \pi]$
Tangent	$\tan(\theta) = a$	$\theta = \tan^{-1}(a) + n\pi$	$(-\frac{\pi}{2}, \frac{\pi}{2})$

Examples

Example 1: Solve $\sin(\theta) = \frac{1}{2}$

Using the general solution for sine, we have:

$\theta = \sin^{-1}(\frac{1}{2}) + 2n\pi$ or $\theta = \pi - \sin^{-1}(\frac{1}{2}) + 2n\pi$

The principal value of $\sin^{-1}(\frac{1}{2})$ is $\frac{\pi}{6}$. Therefore, the general solution is:

$\theta = \frac{\pi}{6} + 2n\pi$ or $\theta = \frac{5\pi}{6} + 2n\pi$, where $n$ is an integer.

Example 2: Solve $\cos(\theta) = -\frac{\sqrt{2}}{2}$

Using the general solution for cosine, we have:

$\theta = \cos^{-1}(-\frac{\sqrt{2}}{2}) + 2n\pi$ or $\theta = -\cos^{-1}(-\frac{\sqrt{2}}{2}) + 2n\pi$

The principal value of $\cos^{-1}(-\frac{\sqrt{2}}{2})$ is $\frac{3\pi}{4}$. Therefore, the general solution is:

$\theta = \frac{3\pi}{4} + 2n\pi$ or $\theta = \frac{7\pi}{4} + 2n\pi$, where $n$ is an integer.

Example 3: Solve $\tan(\theta) = 1$

Using the general solution for tangent, we have:

$\theta = \tan^{-1}(1) + n\pi$

The principal value of $\tan^{-1}(1)$ is $\frac{\pi}{4}$. Therefore, the general solution is:

$\theta = \frac{\pi}{4} + n\pi$, where $n$ is an integer.

Important Points to Remember

• The general solution includes all possible angles that satisfy the equation, represented by the variable $n$ which is an integer.
• The principal value is the smallest angle (in absolute value) that satisfies the equation within the range of the inverse trigonometric function.
• The range of the principal value is important to determine the correct angles that satisfy the equation.
• The period of the trigonometric function determines how often the solutions repeat.

Understanding the general solution of trigonometric equations is crucial for solving problems in trigonometry, especially when dealing with multiple angles and periodic functions. By following the formulas and examples provided, one can effectively find all solutions to trigonometric equations.