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

1. Measuring Angles

Angles in Standard Position

Trigonometry

1. Measuring Angles

Angles in Standard Position

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2:43 minutes

Problem 16

Textbook Question

In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle.31𝜋6

Verified step by step guidance

Step 1: Understand that coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle, you can add or subtract full rotations (360° or 2π radians) from the given angle.

Step 2: Since the given angle is in radians, we will work with 2π. A full rotation in radians is 2π, so we need to find an equivalent angle by subtracting multiples of 2π from the given angle.

Step 3: Start by expressing the given angle, \( \frac{31\pi}{6} \), in terms of 2π. Calculate how many full rotations (2π) fit into \( \frac{31\pi}{6} \).

Step 4: Subtract the integer number of full rotations (2π) from \( \frac{31\pi}{6} \) to find the positive coterminal angle that is less than 2π.

Step 5: Ensure the resulting angle is positive and less than 2π. If necessary, adjust by adding 2π to make it positive.

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Key Concepts

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

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position, differing only by full rotations of 360° or 2π radians. To find a coterminal angle, you can add or subtract multiples of 360° (or 2π) from the given angle. For example, if you have an angle of 30°, adding 360° gives you another coterminal angle of 390°.

Radians and Degrees

Radians and degrees are two units for measuring angles. One full rotation (360°) is equivalent to 2π radians. To convert between these units, you can use the relationships: 180° equals π radians, allowing for easy conversion. Understanding this relationship is crucial when working with angles in trigonometry.

Finding Positive Angles

When tasked with finding a positive angle less than 360° or 2π that is coterminal with a given angle, it is essential to ensure the result is within the specified range. This often involves adding or subtracting 360° or 2π until the angle falls within the desired interval. This process ensures that the angle is both positive and less than the specified limits.