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Chapter 5: Problem 56

find the reference angle for each angle. $$ \frac{17 \pi}{3} $$

Short Answer

Expert verified
The reference angle for \(\frac{17 \pi}{3}\) is \(\frac{\pi}{3}\).

Step by step solution

01

Reduce the Given Angle

Firstly, reduce the given angle \(\frac{17 \pi}{3}\) to an angle between \(0\) and \(2\pi\). We know that \(2\pi\) is a full circle, so subtract or add multiples of \(2\pi\) until the angle is within this range. Here we subtract \(2\pi\) multiples of 5 from \(\frac{17 \pi}{3}\) to get it within the \(0\) to \(2\pi\) range. We get \(\frac{17 \pi}{3} - 5 \times 2\pi = \frac{2 \pi}{3}\)
02

Find the Reference Angle

Now, it depends upon which quadrant the angle \(\frac{2 \pi}{3}\) falls in. In this case, it falls in the second quadrant, and we use \(\pi - \text{angle}\) to calculate the reference angle. Therefore, the reference angle is \(\pi - \frac{2 \pi}{3} = \frac{\pi}{3}\)

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Key Concepts

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

Understanding Angle Reduction
When working with angles, especially in trigonometry, it's essential to reduce larger angles to simpler forms. Consider the angle \(\frac{17 \pi}{3}\). This angle exceeds a full circle, which is \(2\pi\) radians, so it's necessary to bring it within the standard range of \(0\) and \(2\pi\).

To achieve this, we use the method of angle reduction:
  • Subtract or add multiples of \(2\pi\) until the angle falls within a single circle.
  • In our example, \(\frac{17 \pi}{3} - 5 \times 2\pi = \frac{2 \pi}{3}\).
Now the angle \(\frac{2 \pi}{3}\) is between \(0\) and \(2\pi\), making it much easier to work with.
Exploring Quadrants in Trigonometry
Trigonometry divides a circle into four quadrants, each playing a unique role in evaluating angles and their reference angles. After reducing \(\frac{17 \pi}{3}\) to \(\frac{2 \pi}{3}\), we need to determine its position in the circle.

Here's a quick overview of the quadrants:
  • First Quadrant: \(0\) to \(\frac{\pi}{2}\)
  • Second Quadrant: \(\frac{\pi}{2}\) to \(\pi\)
  • Third Quadrant: \(\pi\) to \(\frac{3\pi}{2}\)
  • Fourth Quadrant: \(\frac{3\pi}{2}\) to \(2\pi\)
Since \(\frac{2 \pi}{3}\) lies between \(\frac{\pi}{2}\) and \(\pi\), it is in the second quadrant. Identifying the quadrant is crucial for the next step: finding the reference angle.
Calculating the Trigonometric Reference Angle
The reference angle is a small, positive angle that forms a crucial link between the given angle and the x-axis. It simplifies calculations in trigonometry.

When an angle is found in the second quadrant, like \(\frac{2 \pi}{3}\), the formula for calculating the reference angle is \(\pi - \text{angle}\).

Here's how to apply it:
  • For the angle \(\frac{2 \pi}{3}\), the reference angle is \(\pi - \frac{2 \pi}{3}\).
  • This results in a reference angle of \(\frac{\pi}{3}\).
This reference angle is always positive and less than \(\frac{\pi}{2}\), making trigonometric functions easier to evaluate.

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Most popular questions from this chapter

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \cot \left(\tan ^{-1} \frac{x}{\sqrt{3}}\right) $$

Your neighborhood movie theater has a 25 -foot-high screen located 8 feet above your eye level. If you sit too close to the screen, your viewing angle is too small, resulting in a distorted picture. By contrast, if you sit too far back, the image is quite small, diminishing the movie's visual impact. If you sit \(x\) feet back from the screen, your viewing angle, \(\theta,\) is given by $$\theta=\tan ^{-1} \frac{33}{x}-\tan ^{-1} \frac{8}{x}$$ (GRAPH CANNOT COPY) Find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, and 25 feet.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using radian measure, I can always find a positive angle less than \(2 \pi\) coterminal with a given angle by adding or subtracting \(2 \pi\)

Without drawing a graph, describe the behavior of the graph of \(y=\cos ^{-1} x .\) Mention the function's domain and range in your description.

Without drawing a graph, describe the behavior of the graph of \(y=\tan ^{-1} x .\) Mention the function's domain and range in your description.
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