Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

3. Unit Circle

Reference Angles

Trigonometry

3. Unit Circle

Reference Angles

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2:27 minutes

Problem 49

Textbook Question

Find the reference angle for each angle.
4.7

Verified step by step guidance

Convert the given angle from radians to degrees using the formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

Calculate the equivalent angle in degrees.

Determine the quadrant in which the angle lies by considering the range of angles in degrees for each quadrant.

Find the reference angle by using the appropriate formula based on the quadrant: \( \text{Reference Angle} = 180^\circ - \text{angle} \) for angles in the second quadrant, \( \text{Reference Angle} = \text{angle} - 180^\circ \) for angles in the third quadrant, and \( \text{Reference Angle} = 360^\circ - \text{angle} \) for angles in the fourth quadrant.

Express the reference angle in degrees or convert it back to radians if needed.

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Key Concepts

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

Reference Angle

The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always measured as a positive angle and is crucial for simplifying trigonometric calculations. For angles greater than 180 degrees, the reference angle is found by subtracting 180 degrees from the angle, while for angles in the fourth quadrant, it is calculated by subtracting the angle from 360 degrees.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, each representing a different range of angles. Quadrant I contains angles from 0 to 90 degrees, Quadrant II from 90 to 180 degrees, Quadrant III from 180 to 270 degrees, and Quadrant IV from 270 to 360 degrees. Understanding which quadrant an angle lies in is essential for determining its reference angle and the signs of its trigonometric functions.

Angle Measurement

Angles can be measured in degrees or radians, with 360 degrees equivalent to 2π radians. In trigonometry, it is important to be able to convert between these two units to accurately find reference angles. For example, an angle of 4.7 radians can be converted to degrees to better understand its position in the coordinate plane and subsequently find its reference angle.

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