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Chapter 4: Problem 56

In Exercises \(35-60,\) 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

Convert to an Angle Between 0 and \(2\pi\)

Calculate \(\frac{17 \pi}{3}\) mod \(2\pi\) which is \(\frac{17 \pi}{3}\) mod \(2\pi = 2*\(\frac{17 \pi}{3}\) = \frac{2 \pi} {3}\)
02

Find Reference Angle

As \(\frac{2 \pi} {3}\) is greater than \(\pi/2\) but less than \(\pi\), it lies in the second quadrant. The reference angle \(\theta'\) is equal to \(\pi - \theta\). So, \(\frac{2 \pi} {3}\) subtract from \(\pi\) equals \(\pi - \frac{2 \pi} {3} = \frac{\pi}{3}\).

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Key Concepts

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

Understanding Modulus in Trigonometry
The concept of modulus is pivotal when working with angles in trigonometry, especially when finding reference angles. When you perform calculations like finding the remainder of a division in terms of angles, you are using the modulus operation. It's used to convert a larger angle into a smaller angle within a specified range, commonly between 0 and \(2\pi\) radians (or 0 to 360 degrees).
  • For example, take an angle like \(\frac{17\pi}{3}\). This angle is clearly greater than \(2\pi\) radians, which means it's greater than one full circle.
  • By applying the modulus \(2\pi\), you can find its equivalent angle within a single circle, simplifying further trigonometric processing.
  • This results in \(\frac{17\pi}{3} \mod 2\pi\), which gives a new angle of \(\frac{2\pi}{3}\).
This process allows us to handle angles by confining them within a standard range, making calculations and interpretations much easier.
Angle Conversion Methods
Converting angles from one range to another, such as from degrees to radians or vice versa, is a common necessity in trigonometry.For calculating reference angles, the typical step is to convert an angle to its equivalent less than \(2\pi\) via modulus, and then confirm its part of the circle.
  • To convert degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180}\).
  • Similarly, to go from radians back to degrees, you use \( \text{degrees} = \text{radians} \times \frac{180}{\pi}\).
Easy conversion ensures that angles are manageable within calculation limits and helps in visualizing their positions on the unit circle.Converting an angle like \(\frac{17\pi}{3}\) down to something less than \(2\pi\), such as \(\frac{2\pi}{3}\), makes it easier to determine both its quadrant and its reference angle.
Grasping Trigonometric Quadrants
Understanding which quadrant an angle lies in is crucial to working out its trigonometric functions.The four quadrants of a circle each hold angles with their specific properties:
  • The first quadrant (0 to \(\frac{\pi}{2}\)) contains positive sine and cosine values.
  • The second quadrant (\(\frac{\pi}{2}\) to \(\pi\)) has positive sine but negative cosine values.
  • The third quadrant (\(\pi\) to \(1.5\pi\)) turns both sine and cosine negative.
  • The fourth quadrant (\(1.5\pi\) to \(2\pi\)) has positive cosine but negative sine values.
By understanding quadrant properties, you can determine the sign of trigonometric functions and correctly compute the reference angle.For instance, an angle like \(\frac{2\pi}{3}\) is in the second quadrant. To find its reference angle, subtract it from \(\pi\); such calculations make angles easier to use in equations associated with sine, cosine, and tangent.

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Most popular questions from this chapter

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