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Chapter 8: Problem 57

Use a calculator to solve each equation on the interval \(0 \leq \theta<2 \pi .\) Round answers to two decimal places. \(5 \tan \theta+9=0\)

Short Answer

Expert verified
\(\theta \approx 5.22\) radians

Step by step solution

01

- Isolate the trigonometric function

Start by isolating the \(\tan \theta\) term in the equation \(5 \tan \theta + 9 = 0\). Subtract 9 from both sides to get \(5 \tan \theta = -9\).
02

- Solve for \(\tan \theta\)

Divide both sides of the equation by 5 to isolate \( \tan \theta \). You will get \(\tan \theta = -\frac{9}{5}\).
03

- Find \(\theta\) using the arctangent function

Use a calculator to find \(\theta\). Input \(\theta = \tan^{-1}(-\frac{9}{5})\) into the calculator. Be sure to set the calculator to radians mode. You will find that \(\theta = -1.06\) radians, but this is not within the specified interval.
04

- Adjust result to fit the interval

Adjust the result to fall within the interval \(0 \leq \theta < 2 \pi\) by adding \(\frac{\pi}{2}\) to the angle. We adjust \(\theta = 2\pi + (-1.06) \approx 5.22\) radians.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving trigonometric equations
When solving trigonometric equations, it's important to follow a systematic approach. Start with isolating the trigonometric function. For example, in the equation
\(5 \tan \theta + 9 = 0\),
subtract 9 from both sides to get
\(5 \tan \theta = -9\).
Next, divide by 5 to isolate \(\tan \theta\). You get \(\tan \theta = -\frac{9}{5}\).

Using the arctangent function on your calculator helps find the angle \(\theta\).
Ensure the calculator is in radians mode.
For instance, \(\theta = \tan^{-1}(-\frac{9}{5})\)
gives you \(\theta = -1.06 \text{ radians}\).

Our goal is to adjust this angle to the required interval, \(0 \text{ to } 2\text{蟺}\text{ radians}\).
tangent function
The tangent function, \(\tan \theta\), relates to both sine and cosine functions.

It is defined as \(\frac{\text{sin} \theta}{\text{cos} \theta}\).

The function shows how much a line drawn from the origin to a point on the unit circle slopes compared to the x-axis. This slope can be positive or negative, depending on the quadrant. The range of the tangent function is \((- \text{鈭瀩, \text{鈭瀩)\).

When solving \(\tan \theta\) for a given value, remember that \(\tan \theta\) is periodic, repeating every \(\text{蟺}\).

However, the tangent function has vertical asymptotes where it is undefined: at \(\theta = \frac{\text{蟺}}{2} + k\text{蟺}\), with \(k\) being any integer.
interval adjustment in radians
Adjusting an angle within a specified interval is crucial. When calculating \(\theta\), ensure it falls within the correct bounds.

For instance, if you get \(\theta = -1.06\) radians, it does not fit in \(0 \text{ to } 2\text{蟺}\text{ radians}\).

You add \(2\text{蟺}\) to adjust: \(\theta = 2\text{蟺} + (-1.06) = 5.22 \text{ radians}\).

Adjusting intervals ensures the angle falls within a single cycle of the unit circle. This is crucial for accurate solutions in trigonometry.
arctangent function
The arctangent function, \(\tan^{-1}(x)\), provides the angle whose tangent is \(x\).

On a calculator, this function helps solve for \(\theta\) when \(\tan \theta\) is known.

For instance, \(\tan^{-1}(-\frac{9}{5})\) gives \(\theta = -1.06 \text{ radians}\).

Always check the calculator is set to the correct mode: degrees or radians. For our problem, radians mode is essential.

Remember, the arctangent function returns values typically in the interval \((- \frac{\text{蟺}}{2}, \frac{\text{蟺}}{2})\). Adjust these results as necessary to match your interval requirements.

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