Problem 3 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$\tan \theta=\sqrt{3}$$

Step-by-Step Solution

Answer

The solutions are \( \theta = \frac{\pi}{3} + k\pi \), where \( k \) is any integer.

Step 1: Identify Principal Solution

We start by finding the angle \( \theta \) for which \( \tan \theta = \sqrt{3} \) within the principal value range for tangent, which is \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \). Recall that \( \tan \frac{\pi}{3} = \sqrt{3} \). Therefore, the principal solution is \( \theta = \frac{\pi}{3} \).

Step 2: Determine the Periodicity

The tangent function has a period of \( \pi \). This means that if \( \theta = \frac{\pi}{3} \) is a solution, then any angle of the form \( \theta = \frac{\pi}{3} + k\pi \), where \( k \) is any integer, is also a solution.

Step 3: Write the General Solution

Based on the periodicity, we can write the general solution for the equation \( \tan \theta = \sqrt{3} \) as: \[ \theta = \frac{\pi}{3} + k\pi \] where \( k \) is any integer.

Key Concepts

Understanding the Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It's defined as the ratio of the opposite side to the adjacent side in a right triangle. In the context of a unit circle, \( \tan \theta \) is equal to \( \frac{\sin \theta}{\cos \theta} \).
Key characteristics of the tangent function include:

• It is undefined at angles where \( \cos \theta = 0 \), causing vertical asymptotes, such as at \( \theta = \frac{\pi}{2}, \frac{3\pi}{2},... \).
• The tangent function is periodic and has a pattern repeating every \( \pi \) units.
• The function has no maximum or minimum value as it extends infinitely in both vertical directions.

Understanding these aspects is crucial when solving equations like \( \tan \theta = \sqrt{3} \) as it helps identify potential solutions across different intervals.

Exploring Periodicity and Its Implications

Periodicity in trigonometric functions refers to the repeating nature of these functions over a certain interval. For the tangent function, the repeat interval, or period, is \( \pi \). This means that once you find a particular solution to a tangent equation, adding or subtracting multiples of \( \pi \) will yield additional solutions.
For example, if \( \theta = \frac{\pi}{3} \) is one solution to the equation \( \tan \theta = \sqrt{3} \), then all angles of the form \( \theta = \frac{\pi}{3} + k\pi \) are also solutions, where \( k \) is an integer.

• The periodic nature allows us to easily find all possible solutions across the entire rotation of the circle.
• This simplicity in extending solutions is extremely helpful when dealing with trigonometric equations.

By recognizing periodicity, we make solving trigonometric problems efficient and straightforward.

Navigating the Principal Value Range

The principal value range is a specific interval in which inverse trigonometric functions produce unique outputs. For the tangent function, this range is \(-\frac{\pi}{2} < \theta < \frac{\pi}{2} \). This range is chosen because within it, the tangent function is continuous and covers all possible real outputs.
When solving \( \tan \theta = \sqrt{3} \), our first goal is to find the solution within this principal range. The principal value \( \theta = \frac{\pi}{3} \) neatly fits within \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \), making it a valid starting point.
Some important points to remember include:

• The principal value range ensures a unique solution for inverse trigonometric calculations.
• This range simplifies the task of identifying the primary solution, which can then be expanded based on periodicity.

Understanding and applying the principal value range is essential in solving trigonometric equations accurately and efficiently.