Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ 3-\tan ^{2} \beta=0 $$

Step-by-Step Solution

Answer

\( \beta = \frac{\pi}{3} + n\pi \) and \( \beta = \frac{2\pi}{3} + n\pi \), \( n \in \mathbb{Z} \).

Step 1: Rewrite the Equation

The given equation is \(3 - \tan^2 \beta = 0\). We can rearrange it to express \(\tan^2 \beta\) by moving terms across the equals sign, resulting in \(\tan^2 \beta = 3\).

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

To find \( \tan \beta \), take the square root of both sides. Since we should consider both the positive and negative square roots, we have \( \tan \beta = \sqrt{3} \) or \( \tan \beta = -\sqrt{3} \).

Step 3: Determine the General Solutions

The tangent function is periodic with a period of \( \pi \). Therefore, if \( \tan \beta = \sqrt{3} \), \( \beta = \frac{\pi}{3} + n\pi \), and if \( \tan \beta = -\sqrt{3} \), \( \beta = \frac{2\pi}{3} + n\pi \). Here, \( n \) is any integer.

Step 4: Combine Solutions

Combine the two general forms of solutions from the previous step. The full set of solutions is \( \beta = \frac{\pi}{3} + n\pi \) and \( \beta = \frac{2\pi}{3} + n\pi \), where \( n \) is any integer.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine functions, specifically \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This function can take on any real number value.

• The tangent function has a unique property of being undefined when the cosine of \( \theta \) is zero, which occurs at odd multiples of \( \frac{\pi}{2} \).
• Tangent graphs have repeating patterns, creating a wave-like appearance but with vertical asymptotes where the function is undefined.

The tangent function is also periodic, meaning its values repeat at regular intervals. This characteristic plays a crucial role in solving trigonometric equations, as it helps determine all possible solutions within its periodic cycle.

Square Roots

Square roots are mathematical expressions that represent a quantity which when multiplied by itself gives the original number. For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \). When dealing with equations like \( \tan^2 \beta = 3 \), finding the square root involves considering both the positive and negative roots.

• The symbol for square root is \( \sqrt{\cdot} \). For \( a^2 = 3 \), we write \( a = \sqrt{3} \) or \( a = -\sqrt{3} \).
• In trigonometric terms, this means both \( \tan \beta = \sqrt{3} \) and \( \tan \beta = -\sqrt{3} \) are valid solutions.

Square roots are especially important in trigonometry, as they allow us to solve for angles when squared terms are involved, ensuring all potential solutions are accounted for.

Periodicity

Periodicity in trigonometry refers to the repeating nature of trigonometric functions over specific intervals. For the tangent function, periodicity is defined by a period of \( \pi \). This means that the function repeats its values every \( \pi \) units.

• For example, if \( \tan \beta = \sqrt{3} \) results in an angle \( \beta = \frac{\pi}{3} \), then every \( \pi \) units, this value will recur. Thus, another solution is \( \beta = \frac{\pi}{3} + n\pi \) where \( n \) is an integer.
• The same principle applies for \( \tan \beta = -\sqrt{3} \) at an angle of \( \frac{2\pi}{3} \), making the solution \( \beta = \frac{2\pi}{3} + n\pi \).

Understanding periodicity is crucial when solving trigonometric equations, as it ensures that all possible angle solutions, within the periodic pattern, are found.

General Solutions

The concept of general solutions is essential when working with trigonometric equations. It ensures that all possible angles that satisfy the equation are accounted for. Given the periodic nature of trig functions, such as the tangent function, every possible solution can be expressed using a formula involving an integer multiple of the fundamental period.

• For \( \tan \beta = \sqrt{3} \), the general solution is \( \beta = \frac{\pi}{3} + n\pi \), where \( n \) is any integer representing how many periods are included.
• Similarly, for \( \tan \beta = -\sqrt{3} \), the general solution is \( \beta = \frac{2\pi}{3} + n\pi \).

By using general solutions, you can express infinitely many solutions to equations by referring to these concise formulas, capturing the cyclical nature of trigonometric functions.