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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2:28 minutes

Problem 38

Textbook Question

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree.tan θ = 1.3763819

Verified step by step guidance

Recognize that the tangent function, \( \tan \theta \), is positive in the first and third quadrants.

Use the inverse tangent function to find the reference angle: \( \theta_{ref} = \tan^{-1}(1.3763819) \).

The first angle, \( \theta_1 \), is the reference angle itself, since \( \tan \theta \) is positive in the first quadrant.

The second angle, \( \theta_2 \), is found by adding 180° to the reference angle, since \( \tan \theta \) is also positive in the third quadrant.

Round both angles, \( \theta_1 \) and \( \theta_2 \), to the nearest degree to find the final solutions.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The tangent function, specifically, is defined as the ratio of the opposite side to the adjacent side. Understanding how these functions behave and their values at specific angles is crucial for solving trigonometric equations.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arctan, are used to find angles when given a trigonometric ratio. For example, if tan θ = 1.3763819, we can use the arctan function to determine the angle θ. It's important to consider the range of these functions and how they relate to the unit circle when finding angles.

Angle Solutions in Different Quadrants

Trigonometric functions are periodic, meaning they repeat their values in different quadrants of the unit circle. For tangent, which is positive in the first and third quadrants, it is essential to find all possible angles that satisfy the equation within the specified interval [0°, 360°). This involves understanding the properties of angles and their corresponding trigonometric values.