Problem 22 - Algebra and Trigonometry

Question

\(17-24\) n Solve the given equation, and list six specific solutions. $$ \tan \theta=2.5 $$

Step-by-Step Solution

Answer

The specific solutions are approximately \( \theta \approx 1.1903, 4.3319, 7.4731, -1.9513, -5.0925, \) and \( -8.2337 \) radians.

Step 1: Understand the Tangent Function

The tangent function is periodic with a period of \( \pi \). Thus, the equation \( \tan \theta = 2.5 \) has infinitely many solutions because of this periodicity.

Step 2: Find the Principal Solution

Use the inverse tangent function to find the principal value, \( \theta = \arctan(2.5) \). Compute this using a calculator to obtain \( \theta \approx 1.1903 \) radians.

Step 3: Determine the General Solution

The general solution for \( \tan \theta = 2.5 \) is given by \( \theta = \arctan(2.5) + n\pi \), where \( n \) is any integer. This accounts for the periodic nature of the tangent function.

Step 4: Choose the Integer Values for n

Select six consecutive integers for \( n \) to find specific solutions. We will use \( n = 0, 1, 2, -1, -2, -3 \).

Step 5: Calculate Specific Solutions

Substitute each chosen integer into the general solution to find specific solutions: - For \( n = 0 \), \( \theta \approx 1.1903 \)- For \( n = 1 \), \( \theta \approx 1.1903 + \pi \approx 4.3319 \)- For \( n = 2 \), \( \theta \approx 1.1903 + 2\pi \approx 7.4731 \)- For \( n = -1 \), \( \theta \approx 1.1903 - \pi \approx -1.9513 \)- For \( n = -2 \), \( \theta \approx 1.1903 - 2\pi \approx -5.0925 \)- For \( n = -3 \), \( \theta \approx 1.1903 - 3\pi \approx -8.2337 \)

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the primary trigonometric functions alongside sine and cosine. It can be defined in a right-angled triangle as the ratio of the side opposite the angle \( \theta \) to the side adjacent to \( \theta \). Alternatively, in terms of a unit circle, the tangent of an angle is the y-coordinate divided by the x-coordinate. This function is particularly important because it captures the slope of an angle in a periodic interval, which becomes useful in various mathematical applications.

• The tangent function repeats itself every \( \pi \) radians, which contributes to its periodic nature.
• It has asymptotes at odd multiples of \( \frac{\pi}{2} \), where its values reach infinity.
• This function is undefined at \( \pi/2, 3\pi/2, \ldots \), leading to vertical asymptotes at these points.

Understanding the tangent function's behavior and properties allows for solving equations like \( \tan \theta = 2.5 \), where we need to find angles that yield a tangent value of 2.5.

Inverse Tangent

The inverse tangent function, represented as \( \arctan \) or \( \tan^{-1} \), is used to determine the angle whose tangent is a given number. Rather than giving the slope, it gives the angle itself from the ratio presented. For example, if \( \tan \theta = 2.5 \), using \( \arctan \) helps to find \( \theta \).

• It returns values typically in the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
• This range is known as the principal value range, and it's the standard output for calculators and other computational tools.
• Once you have your principal value \( \theta \approx 1.1903 \) radians, it serves as the basis of the general solution.

Utilizing the inverse tangent function allows us to find a starting point, or principal solution, around which we can construct the full set of solutions using tangent's periodicity.

Periodicity

Periodicity in trigonometric functions refers to the repeating pattern established over specific intervals. For the tangent function, this period is \( \pi \). This means after adding \( \pi \) to any solution \( \theta \), the tangent value remains the same.

• The formula for all solutions of the equation \( \tan \theta = 2.5 \) is \( \theta = \arctan(2.5) + n\pi \), where \( n \) is an integer.
• Each addition of \( \pi \) jumps to the next cycle of equivalent tangent values, whether positive or negative.
• This cyclical nature provides a systematic way to list multiple solutions.

Finding six specific solutions, given as examples \( \theta \approx 1.1903, 4.3319, 7.4731, -1.9513, -5.0925, -8.2337 \), demonstrates the use of periodicity. It allows for the exploration of angle possibilities by shifting forward or backward in cycles of \( \pi \). Periodicity is crucial for the comprehensive understanding of trigonometric equations and their infinite nature of solutions.