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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 48
Chapter 1, Problem 48

In Exercises 44–48, find the reference angle for each angle.
- 11πœ‹/3

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 between 0 and \( \frac{\pi}{2} \).
Since the given angle is \( \frac{11\pi}{3} \), which is greater than \( 2\pi \), we need to find its equivalent angle between 0 and \( 2\pi \) by subtracting multiples of \( 2\pi \). Use the formula: \( \theta_{equiv} = \theta - 2\pi \times k \), where \( k \) is an integer chosen so that \( \theta_{equiv} \) lies in \( [0, 2\pi) \).
Calculate \( k \) such that \( \frac{11\pi}{3} - 2\pi k \) is between 0 and \( 2\pi \). Since \( 2\pi = \frac{6\pi}{3} \), subtract \( 2\pi \) multiples accordingly.
Once you find the equivalent angle \( \theta_{equiv} \), determine which quadrant it lies in by comparing it to \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).
Finally, find the reference angle based on the quadrant: - Quadrant I: reference angle = \( \theta_{equiv} \) - Quadrant II: reference angle = \( \pi - \theta_{equiv} \) - Quadrant III: reference angle = \( \theta_{equiv} - \pi \) - Quadrant IV: reference angle = \( 2\pi - \theta_{equiv} \)

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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 90Β°, used to simplify trigonometric calculations by relating any angle to an acute angle in the first quadrant.
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Reference Angles on the Unit Circle

Angle Reduction Using Coterminal Angles

Coterminal angles differ by full rotations of 2Ο€ radians (360Β°). To find a reference angle for large angles, first reduce the angle by subtracting multiples of 2Ο€ until it lies between 0 and 2Ο€, making it easier to analyze its position in the unit circle.
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Coterminal Angles

Quadrants and Sign of Angles

The position of an angle in the coordinate plane (quadrants I-IV) determines how to calculate its reference angle. Knowing the quadrant helps identify whether to subtract the angle from Ο€, 2Ο€, or use the angle directly to find the acute reference angle.
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Quadratic Formula
Related Practice
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.

-5πœ‹/4

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Textbook Question

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.

cot πœ‹/12

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Textbook Question

In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.

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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.

-7πœ‹/4

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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.

16πœ‹/3

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Textbook Question

In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.

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