Trigonometric Equations: Solution of Trigonometric Equations - Principal Values

Trigonometric equations are mathematical statements that involve trigonometric functions such as sine, cosine, tangent, etc. These equations can often have multiple solutions because trigonometric functions are periodic. However, the principal values are the smallest set of solutions within one period of the function that can represent all the solutions.

Understanding Trigonometric Functions

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

• Sine (sin): A periodic function with a period of $2\pi$.
• Cosine (cos): Also periodic with a period of $2\pi$.
• Tangent (tan): Periodic with a period of $\pi$.

The principal values for the inverse trigonometric functions are defined as follows:

• Arcsine (asin or $\sin^{-1}$): $[-\frac{\pi}{2}, \frac{\pi}{2}]$
• Arccosine (acos or $\cos^{-1}$): $[0, \pi]$
• Arctangent (atan or $\tan^{-1}$): $(-\frac{\pi}{2}, \frac{\pi}{2})$

Solving Trigonometric Equations

When solving trigonometric equations, we often look for all solutions that satisfy the equation. However, we can use the principal values to express all solutions in terms of a finite number of angles.

General Solutions

The general solution of a trigonometric equation involves finding a set of solutions that can be expressed in terms of an integer multiple of the period of the trigonometric function. For example, the general solution for a sine equation $sin(x) = a$ can be written as:

$$ x = \sin^{-1}(a) + 2n\pi \quad \text{or} \quad x = \pi - \sin^{-1}(a) + 2n\pi, \quad n \in \mathbb{Z} $$

Principal Values

The principal value is the unique solution within the defined range of the inverse trigonometric function. For the equation $sin(x) = a$, the principal value is simply $\sin^{-1}(a)$, assuming that $a$ is within the range of the sine function, i.e., $-1 \leq a \leq 1$.

Table of Differences and Important Points

Feature	General Solution	Principal Value
Range	All possible solutions	Restricted to a specific interval
Periodicity	Takes into account the periodic nature of trig functions	Represents one cycle of the trig function
Inverse Functions	Not necessarily used	Always used to find the principal value
Representation	Infinite solutions (due to periodicity)	A single solution within the defined range

Formulas

Here are some formulas for finding the general solutions of basic trigonometric equations:

• Sine: $x = \sin^{-1}(a) + 2n\pi \quad \text{or} \quad x = \pi - \sin^{-1}(a) + 2n\pi$
• Cosine: $x = \cos^{-1}(a) + 2n\pi \quad \text{or} \quad x = -\cos^{-1}(a) + 2n\pi$
• Tangent: $x = \tan^{-1}(a) + n\pi$

Examples

Let's go through some examples to illustrate the important points.

Example 1: Sine Equation

Solve for $x$ in the equation $\sin(x) = \frac{1}{2}$.

General Solution:

$$ x = \sin^{-1}\left(\frac{1}{2}\right) + 2n\pi \quad \text{or} \quad x = \pi - \sin^{-1}\left(\frac{1}{2}\right) + 2n\pi $$

Principal Value:

$$ x = \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} $$

Example 2: Cosine Equation

Solve for $x$ in the equation $\cos(x) = -\frac{\sqrt{2}}{2}$.

General Solution:

$$ x = \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) + 2n\pi \quad \text{or} \quad x = -\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) + 2n\pi $$

Principal Value:

$$ x = \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} $$

Example 3: Tangent Equation

Solve for $x$ in the equation $\tan(x) = 1$.

General Solution:

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

Principal Value:

$$ x = \tan^{-1}(1) = \frac{\pi}{4} $$

In summary, when solving trigonometric equations, it's important to distinguish between the general solution, which accounts for the periodic nature of trigonometric functions, and the principal value, which is the unique solution within a specific interval. Understanding these concepts is crucial for solving trigonometric equations in mathematics.