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 1.3.55

Chapter 1, Problem 1.3.55

Find the reference angle for each angle.
23π/4

Verified step by step guidance

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 \( \frac{23\pi}{4} \), which is greater than \( 2\pi \), we need to find its equivalent angle between 0 and \( 2\pi \) by subtracting multiples of \( 2\pi \).

Calculate the equivalent angle by subtracting \( 2\pi = \frac{8\pi}{4} \) repeatedly from \( \frac{23\pi}{4} \) until the result is between 0 and \( 2\pi \). This can be expressed as \( \frac{23\pi}{4} - n \times 2\pi \) where \( n \) is an integer.

Once you find the equivalent angle \( \theta \) in the interval \( [0, 2\pi) \), determine which quadrant \( \theta \) lies in to find the reference angle.

Use the quadrant information to calculate the reference angle \( \alpha \) as follows: if \( \theta \) is in Quadrant I, \( \alpha = \theta \); Quadrant II, \( \alpha = \pi - \theta \); Quadrant III, \( \alpha = \theta - \pi \); Quadrant IV, \( \alpha = 2\pi - \theta \).

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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 angles to their acute counterparts.

Angle Reduction Using Coterminal Angles

Coterminal angles differ by full rotations of 2π radians (360°). To find a reference angle for large angles like 23π/4, reduce the angle by subtracting multiples of 2π until it lies within one full rotation (0 to 2π).

Quadrant Identification

Determining the quadrant where the angle's terminal side lies is essential because the reference angle depends on the quadrant. Each quadrant has a specific way to calculate the reference angle based on the angle's position relative to the x-axis.

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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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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.

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