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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2:53 minutes

Problem 42

Textbook Question

Find the reference angle for each angle.
5π/4

Verified step by step guidance

Step 1: Understand that the reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.

Step 2: Convert the given angle from radians to degrees if necessary. However, since the problem is in radians, we will work with radians directly.

Step 3: Recognize that the angle \( \frac{5\pi}{4} \) is in the third quadrant because it is greater than \( \pi \) (which is \( 180^\circ \)) and less than \( \frac{3\pi}{2} \) (which is \( 270^\circ \)).

Step 4: To find the reference angle for an angle in the third quadrant, subtract \( \pi \) from the given angle. So, calculate \( \frac{5\pi}{4} - \pi \).

Step 5: Simplify the expression from Step 4 to find the reference angle in radians.

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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 used to simplify the calculation of trigonometric functions. For angles greater than 180 degrees, the reference angle is found by subtracting the angle from 180 degrees or 360 degrees, depending on the quadrant in which the angle lies.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each representing a range of angles. The first quadrant (0 to π/2) contains angles where both sine and cosine are positive. The second quadrant (π/2 to π) has positive sine and negative cosine, the third quadrant (π to 3π/2) has both sine and cosine negative, and the fourth quadrant (3π/2 to 2π) has positive cosine and negative sine. Understanding these quadrants is essential for determining the reference angle.

Angle Measurement in Radians

Angles can be measured in degrees or radians, with radians being the standard unit in trigonometry. One full rotation (360 degrees) is equivalent to 2π radians. To convert degrees to radians, multiply by π/180. When working with angles like 5π/4, recognizing that this angle is in radians helps in accurately determining its position on the unit circle and subsequently finding the reference angle.