Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problema 1.3.59

Capitolo 1, Problema 1.3.59

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

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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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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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Percorso guidato

5:31

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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Percorso guidato

04:46

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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Percorso guidato

6:36

Quadratic Formula

Pratica correlata

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