Problem 26 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$3-\tan ^{2} \beta=0$$

Step-by-Step Solution

Answer

The solutions are \( \beta = \frac{\pi}{3} + k\pi \) and \( \beta = -\frac{\pi}{3} + k\pi \).

Step 1: Rearrange the Equation

Start with the provided equation: \[ 3 - \tan^2 \beta = 0 \]To isolate \( \tan^2 \beta \), rearrange the equation by adding \( \tan^2 \beta \) to both sides. This results in the equation:\[ \tan^2 \beta = 3 \]

Step 2: Solve for \( \tan \beta \)

Take the square root of both sides of the equation to solve for \( \tan \beta \):\[ \tan \beta = \pm \sqrt{3} \]This indicates that \( \tan \beta \) can be either \( \sqrt{3} \) or \( -\sqrt{3} \).

Step 3: Identify General Solutions for \( \tan \beta = \sqrt{3} \)

The tan function equals \( \sqrt{3} \) at specific angles within the unit circle. The primary angle is:\[ \beta = \frac{\pi}{3} + k\pi \]where \( k \) is any integer. This is because the tangent function has a period of \( \pi \).

Step 4: Identify General Solutions for \( \tan \beta = -\sqrt{3} \)

The tan function equals \( -\sqrt{3} \) at specific angles:\[ \beta = -\frac{\pi}{3} + k\pi \]where \( k \) is any integer. The periodicity of \( \tan \) allows these angles.

Key Concepts

Tangent Function

The tangent function has distinct characteristics among trigonometric functions due to its unique relationship with sine and cosine.

• The formula for tangent of an angle is given by \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• It is undefined where the cosine function is zero, making it distinctive at angles like \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).

One fascinating aspect of the tangent function is its ability to take on all real values, unlike sine and cosine, which are bounded between -1 and 1. This characteristic makes the tangent function particularly useful for solving equations where a wide range of solutions is necessary. Such is the case in trigonometric equations, where knowing the possible values that \( \tan \beta \) can take, such as \( \pm \sqrt{3} \), helps in evaluating angles based on the unit circle.

Unit Circle

The unit circle is essential for understanding trigonometric equations like the one given. Visualizing the angles that satisfy \( \tan \beta = \sqrt{3} \) or \( \tan \beta = -\sqrt{3} \) becomes straightforward with the unit circle.

• The unit circle is a circle with radius 1, centered at the origin of the coordinate plane.
• Angles on the unit circle can be expressed in radians, which are critical in calculating trigonometric functions.

For example, consider the equation \( \tan \beta = \sqrt{3} \). On the unit circle, this occurs where the y-coordinate (sine) divided by the x-coordinate (cosine) equals \( \sqrt{3} \), specifically at \( \frac{\pi}{3} \). Similarly, for \( \tan \beta = -\sqrt{3} \), the angles at which this holds true are at \(-\frac{\pi}{3}\). With the unit circle, imagining the periodic and cyclic nature of trigonometric functions becomes intuitive. This helps in identifying multiple solutions, considering their periodicity.

Periodicity of Trigonometric Functions

The periodicity of the tangent function is what allows us to generate an infinite number of solutions to trigonometric equations. Understanding periodicity is crucial in predicting the behavior of functions beyond their initial cycles.

• The period of the tangent function is \( \pi \), meaning it repeats every \( \pi \) radians.
• This repetition is evident in the general solutions of equations like \( 3 - \tan^2 \beta = 0 \).

When finding solutions to \( \tan \beta = \sqrt{3} \), we add multiples of \( \pi \) to the initial angle \( \frac{\pi}{3} \) to account for its periodicity, resulting in the general solution \( \beta = \frac{\pi}{3} + k\pi \). Similarly, for \( \tan \beta = -\sqrt{3} \), the solution is \( \beta = -\frac{\pi}{3} + k\pi \).This characteristic also highlights why knowledge of periodicity is a vital component in solving more complex trigonometric problems, seamlessly predicting the angles where specific tangent values recur.