Trigonometric Equations: Principal & General Solution

Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, tangent, etc. Solving these equations often involves finding angles that satisfy the given trigonometric relationship. There are two types of solutions for trigonometric equations: the principal solution and the general solution.

Principal Solution

The principal solution of a trigonometric equation refers to the smallest non-negative angle (measured in radians or degrees) that satisfies the equation. This solution is typically within a specific range, depending on the trigonometric function involved.

For example, for the sine and cosine functions, the principal solution is usually sought within the interval $[0, 2\pi)$ or $[0, 360^\circ)$, and for the tangent function, within the interval $[-\frac{\pi}{2}, \frac{\pi}{2})$ or $[-90^\circ, 90^\circ)$.

General Solution

The general solution takes into account the periodic nature of trigonometric functions and includes all possible angles that satisfy the equation. This means that the general solution is the set of all angles that can be obtained by adding integer multiples of the period to the principal solution(s).

For sine and cosine, the period is $2\pi$ (or $360^\circ$), and for tangent, the period is $\pi$ (or $180^\circ$).

Formulas for General Solutions

Here are the general solution formulas for the basic trigonometric functions:

• Sine: If $\sin \theta = \sin \alpha$, then the general solution is $\theta = n\pi + (-1)^n \alpha$, where $n$ is an integer.
• Cosine: If $\cos \theta = \cos \alpha$, then the general solution is $\theta = 2n\pi \pm \alpha$, where $n$ is an integer.
• Tangent: If $\tan \theta = \tan \alpha$, then the general solution is $\theta = n\pi + \alpha$, where $n$ is an integer.

Differences and Important Points

Aspect	Principal Solution	General Solution
Definition	The smallest non-negative angle that satisfies the equation.	All angles that satisfy the equation, including the principal solution and all other solutions obtained by adding integer multiples of the period.
Range	Limited to a specific interval.	Infinite, includes all possible solutions.
Periodicity	Not applicable, as it refers to a single solution.	Takes into account the periodic nature of trigonometric functions.
Use	Often used when a single solution is required.	Used when all possible solutions are needed.
Example Functions	$\sin \theta$, $\cos \theta$, $\tan \theta$	$\sin \theta$, $\cos \theta$, $\tan \theta$

Examples

Example 1: Principal Solution

Find the principal solution of the equation $\sin \theta = \frac{\sqrt{2}}{2}$.

The principal solution is the smallest angle in the interval $[0, 2\pi)$ for which the sine function equals $\frac{\sqrt{2}}{2}$. This occurs at $\theta = \frac{\pi}{4}$ and $\theta = \frac{3\pi}{4}$.

Example 2: General Solution

Find the general solution of the equation $\sin \theta = \frac{\sqrt{2}}{2}$.

Using the formula for the general solution of the sine function, we have:

$$ \theta = n\pi + (-1)^n \left(\frac{\pi}{4}\right) $$

where $n$ is an integer. This means that the general solution includes all angles of the form $\frac{\pi}{4} + n\pi$ and $\frac{3\pi}{4} + n\pi$, where $n$ is any integer.

Example 3: Applying General Solution to a Specific Interval

Find all solutions of the equation $\cos \theta = -\frac{1}{2}$ in the interval $[0, 2\pi)$.

First, find the principal solution. The cosine function equals $-\frac{1}{2}$ at $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$ within the interval $[0, 2\pi)$.

Now, apply the general solution formula:

$$ \theta = 2n\pi \pm \frac{2\pi}{3} $$

Since we are only interested in solutions within $[0, 2\pi)$, we find that the solutions are $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$, as these are the only values that fall within the specified interval when $n = 0$.

Understanding the difference between principal and general solutions is crucial for solving trigonometric equations effectively. The principal solution provides a specific angle, while the general solution encompasses all angles that satisfy the trigonometric equation, considering the periodic nature of these functions.