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

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

Trigonometry

2. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

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1: minutes

Problem 8

Textbook Question

CONCEPT PREVIEW Determine whether each statement is possibleor impossible. cos θ = 1.5

Verified step by step guidance

Understand the range of the cosine function: The cosine of an angle \( \theta \) in a right triangle or on the unit circle is defined as the ratio of the adjacent side to the hypotenuse, or as the x-coordinate of a point on the unit circle.

Recall the range of the cosine function: The cosine function, \( \cos \theta \), has a range of \([-1, 1]\). This means that for any angle \( \theta \), the value of \( \cos \theta \) must be between -1 and 1, inclusive.

Evaluate the given statement: The statement \( \cos \theta = 1.5 \) suggests that the cosine value is 1.5, which is outside the range of possible values for the cosine function.

Determine the possibility: Since 1.5 is not within the range \([-1, 1]\), it is impossible for \( \cos \theta \) to equal 1.5.

Conclude: Therefore, the statement \( \cos \theta = 1.5 \) is impossible.

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Key Concepts

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

Range of Cosine Function

The cosine function, denoted as cos(θ), outputs values strictly within the range of -1 to 1 for any angle θ. This means that any value outside this interval, such as 1.5, is not possible for the cosine of an angle. Understanding this range is crucial for determining the validity of statements involving cosine.

Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles. The primary functions include sine, cosine, and tangent, each defined based on the relationships between the angles and sides. Recognizing how these functions operate helps in evaluating statements about their possible values.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the values of sine and cosine for all angles. It is a circle with a radius of one centered at the origin of a coordinate plane. The x-coordinate of a point on the unit circle represents the cosine of the angle, reinforcing that cosine values must lie between -1 and 1.