Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 1.3.53

Chapter 1, Problem 1.3.53

In Exercises 35–60, find the reference angle for each angle. 17𝜋 / 6

Verified step by step guidance

Identify the given angle: \(\frac{17\pi}{6}\) radians.

Since the angle is greater than \(2\pi\), subtract \(2\pi\) (which is \(\frac{12\pi}{6}\)) to find the equivalent angle between \(0\) and \(2\pi\): \(\frac{17\pi}{6} - 2\pi = \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6}\).

Determine the quadrant of the angle \(\frac{5\pi}{6}\). Since \(\frac{5\pi}{6}\) is between \(\frac{\pi}{2}\) and \(\pi\), it lies in the second quadrant.

For angles in the second quadrant, the reference angle \(\theta_r\) is calculated as \(\pi - \theta\). So, compute \(\theta_r = \pi - \frac{5\pi}{6} = \frac{\pi}{6}\).

Thus, the reference angle for \(\frac{17\pi}{6}\) is \(\frac{\pi}{6}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reference Angle

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It is always positive and less than or equal to 90°, used to simplify trigonometric calculations by relating any angle to a corresponding acute angle.

Angle Measurement in Radians

Angles can be measured in radians, where 2π radians equal 360 degrees. Understanding how to convert and interpret angles in radians is essential, especially for angles greater than 2π or negative angles, to find their equivalent positions on the unit circle.

Unit Circle and Coterminal Angles

The unit circle helps visualize angles and their positions. Coterminal angles differ by full rotations of 2π radians but share the same terminal side. Finding coterminal angles within one rotation simplifies determining the reference angle.

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Coterminal Angles on the Unit Circle

Related Practice

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In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan(-𝜋/4)

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In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -760°

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In Exercises 63–68, find the exact value of each expression. Do not use a calculator. 1 + sin² 40° + sin² 50°

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In Exercises 63–68, find the exact value of each expression. Do not use a calculator. tan(𝜋/3)/2 - 1/sec(𝜋/6)

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In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

120°

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Textbook Question

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 395°

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