Problem 14 - Precalculus Mathematics for Calculus

Question

Solve the given equation. $$\tan \theta=1$$

Step-by-Step Solution

Answer

The solutions are \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is any integer.

Step 1: Understand the Function

The function given is \( \tan \theta = 1 \). The tangent function is periodic with period \( \pi \), meaning it repeats its values every \( \pi \) radians.

Step 2: Identify the Angle with Specific Value

Recall that \( \tan \theta = 1 \) specifically at \( \theta = \frac{\pi}{4} \), since tangent represents the ratio of sine to cosine, and both \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 3: General Solution Determination

Since the tangent function has a period of \( \pi \), the general solution for \( \tan \theta = 1 \) is \( \theta = \frac{\pi}{4} + n\pi \), where \( n \) is any integer.

Step 4: Conclude with General Solution

Thus, the solutions for the equation \( \tan \theta = 1 \) can be written as \( \theta = \frac{\pi}{4} + n\pi \), considering all possible integer values of \( n \). This accounts for the periodic nature of the tangent function.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function that describes the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as the ratio of the sine function to the cosine function. This means:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function is unique because it does not have a maximum or minimum value; instead, it increases indefinitely as it approaches odd multiples of \( \frac{\pi}{2} \). This results in vertical asymptotes at these values, where the function is undefined.
Unlike sine and cosine, which have a period of \( 2\pi \), the tangent function repeats every \( \pi \) radians. This "shorter" period means it completes one cycle much faster, which is useful for solving equations involving tangent since fewer calculations are needed to understand its behavior within any interval.

Periodicity

Understanding periodicity helps solve trigonometric equations, especially those involving the tangent function. Periodicity means that the function repeats its values in regular intervals. For the tangent function, the period is \( \pi \). This implies:

• \( \tan(\theta + n\pi) = \tan \theta \)

This property is crucial because it tells us that if we find a solution for certain values of \( \theta \), we can determine more solutions by adding or subtracting integer multiples of the period (\( \pi \)).
In the context of solving \( \tan \theta = 1 \), knowing this periodicity allows us to express the general solution compactly. By finding one angle where \( \tan \theta = 1 \), specifically at \( \theta = \frac{\pi}{4} \), we can generate an infinite number of solutions by utilizing this periodic nature:
\[ \theta = \frac{\pi}{4} + n\pi \] where \( n \) is any integer.

General Solution

When solving trigonometric equations, especially those involving periodic functions like the tangent function, it is important to find a general solution. A general solution encompasses all possible solutions to the equation for all periods.
Given \( \tan \theta = 1 \):

• We first identify a specific solution, \( \theta = \frac{\pi}{4} \), because \( \tan(\frac{\pi}{4}) = 1 \).
• Using the periodicity of the tangent function, we recognize that this solution repeats every \( \pi \) radians.

Thus, the general solution for \( \tan \theta = 1 \) is expressed as:
\[ \theta = \frac{\pi}{4} + n\pi \] where \( n \) is any integer. This formula represents every solution that satisfies the equation, by accounting for the infinite repetitions that arise from the function's periodicity.