Problem 23 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 3 \tan \theta+5=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(3\tan(\theta) + 5 = 0\), for \(0 \leq \theta < 2 \pi\), are \( \theta = \tan^{-1}(-5/3)\) and \( \theta = \tan^{-1}(-5/3) + \pi\).

Step 1: Isolate the Tangent Function

The given equation is \(3\tan(\theta) + 5 = 0\). The first step is to isolate \(\tan(\theta)\). Subtract 5 from both sides to get: \(3\tan(\theta) = -5\). Then, divide all terms by 3 to isolate \(\tan(\theta)\): \(\tan(\theta) = -5/3\).

Step 2: Solve for \( \theta \)

To find \( \theta \), we use the inverse tangent function (also called arctangent or \(\tan^{-1}\)). So, \( \theta = \tan^{-1}(-5/3)\). However, this will only give us one solution, and there might be multiple solutions within the given interval \([0, 2\pi)\). Remember, the tangent function has period \(\pi\) which means it repeats every \(\pi\) units.

Step 3: Determine the Complete Set of Possible Solutions

We know that \(\theta = \tan^{-1}(-5/3)\) is one possible solution. The next angle which also has tangent equal to \(-5/3\) can be found by adding \(\pi\) to the current angle because of the periodicity of the tangent function. Therefore, add \(\pi\) to the initial solution to get another value of \( \theta \) within the interval \([0, 2\pi)\).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is one of the fundamental trigonometric functions, just like sine and cosine. It is defined in a right-angled triangle as the ratio of the opposite side to the adjacent side, relative to the angle \( \theta \). In terms of the unit circle, it can also be understood as the ratio of the y-coordinate to the x-coordinate for a given angle.

• The tangent function is unique because it can take any value from negative infinity to positive infinity, unlike sine and cosine, which are bound between -1 and 1.
• The graph of the tangent function is a series of repeating curves that head towards infinity, separated by vertical asymptotes where the function is undefined, specifically at \( \theta = \frac{\pi}{2} + k\pi \) for any integer \( k \).

Understanding the behavior of \( \tan(\theta) \) is crucial when solving equations involving the tangent. In such tasks, it's typical to first isolate \( \tan(\theta) \), as seen in the original exercise. Once isolated, you can proceed to find the solutions by considering the tangent's characteristics, like its potential infinite range of values.

Inverse Tangent

The inverse tangent, often denoted as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \), is the function used to find the angle whose tangent is a given number. This function helps us find angles when we know the tangent value. It is essential for solving trigonometric equations when you need to determine \( \theta \) from equations like \( \tan(\theta) = x \).

• The inverse tangent function yields values in the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). These are the primary values that the function returns.
• When using \( \tan^{-1} \), remember that it only gives one possible solution – the principal value.

In our case, when you find \( \theta = \tan^{-1}(-5/3) \), this solution is just one spot on the unit circle. Given the periodic nature of the tangent function, we'll find more solutions by considering its periodicity. This becomes crucial when we're seeking all possible solutions within a specified interval.

Periodicity of Functions

The periodicity of functions is a key concept in trigonometry, indicating that the function repeats its values at regular intervals. For the tangent function, it repeats every \( \pi \) radians. This periodic behavior allows us to predict the function’s values beyond their initial calculated solutions.

• The tangent function, \( \tan(\theta) \), specifically, is periodic with a period of \( \pi \), meaning that for any solution \( \theta \), \( \theta + k\pi \) (where \( k \) is an integer) is also a solution.
• This characteristic makes it easier to find all solutions over an interval by simply adding \( \pi \) to a known solution to find another valid solution until you cover the entire interval.

In our problem-solving process, after finding the first angle \( \theta \) from \( \tan^{-1}(-5/3) \), we consider periodicity. By adding \( \pi \), we find additional solutions, ensuring we identify all answers within \( [0, 2\pi) \). Always check to ensure these computed solutions are within the required range, adjusting by adding or subtracting \( 2\pi \) if necessary.