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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblem 1.3.59
Chapter 1, Problem 1.3.59

Find the reference angle for each angle.
-25π/6

Verified step by step guidance
1
First, understand that the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It is always positive and between 0 and \( \frac{\pi}{2} \).
Since the given angle is \( -\frac{25\pi}{6} \), start by finding a coterminal angle between 0 and \( 2\pi \) by adding multiples of \( 2\pi \) until the angle is positive and within one full rotation. Use the formula: \( \theta_{coterminal} = \theta + 2\pi k \), where \( k \) is an integer.
Calculate \( k \) such that \( -\frac{25\pi}{6} + 2\pi k \) lies between 0 and \( 2\pi \). Since \( 2\pi = \frac{12\pi}{6} \), add \( 2\pi \) multiples accordingly.
Once you find the positive coterminal angle \( \theta_{coterminal} \), determine which quadrant it lies in by comparing it to \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).
Finally, find the reference angle by calculating the acute angle between \( \theta_{coterminal} \) and the nearest x-axis boundary (0, \( \pi \), or \( 2\pi \)) depending on the quadrant.

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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 lies between 0 and π/2 radians (0° and 90°). Reference angles help simplify trigonometric calculations by relating any angle to a corresponding acute angle.
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Reference Angles on the Unit Circle

Angle Coterminality and Standard Position

Angles are coterminal if they differ by full rotations of 2π radians (360°). To find a reference angle, first convert the given angle to an equivalent angle between 0 and 2π by adding or subtracting multiples of 2π. This places the angle in standard position, making it easier to identify the reference angle.
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Coterminal Angles

Quadrant Identification

The quadrant in which the terminal side of the angle lies determines how to calculate the reference angle. Each quadrant has a specific formula for the reference angle based on the angle's position relative to the x-axis. Knowing the quadrant helps correctly find the acute angle between the terminal side and the x-axis.
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Related Practice
Textbook Question

A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

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Textbook Question

Convert 135° to an exact radian measure.

Textbook Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sec 240°

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Textbook Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. csc(7𝜋/6)

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In Exercises 35–60, find the reference angle for each angle. 553°

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Textbook Question

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.

420°

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