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

Previous problemNext problem

2:30 minutes

Problem 103

Textbook Question

Concept CheckSuppose that ―90° < θ < 90° .Find the sign of each function value. cos(θ―180°)

Verified step by step guidance

Recognize that the problem involves finding the sign of the cosine function for an angle transformation.

Use the identity for cosine of a shifted angle: \( \cos(\theta - 180^\circ) = -\cos(\theta) \).

Since \( -90^\circ < \theta < 90^\circ \), \( \theta \) is in the first or fourth quadrant.

In the first quadrant, \( \cos(\theta) > 0 \), and in the fourth quadrant, \( \cos(\theta) > 0 \) as well.

Therefore, \( -\cos(\theta) < 0 \) for both quadrants, meaning \( \cos(\theta - 180^\circ) \) is negative.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Understanding Angle Measures

In trigonometry, angles can be measured in degrees or radians. The question specifies that θ is between -90° and 90°, which indicates that θ is in the first or fourth quadrant. This range is crucial for determining the signs of trigonometric functions, as the sign of a function can vary depending on the quadrant in which the angle lies.

Cosine Function and Its Properties

The cosine function, denoted as cos(θ), represents the x-coordinate of a point on the unit circle corresponding to the angle θ. The cosine function is positive in the first quadrant and negative in the second quadrant. Understanding how the cosine function behaves with respect to angle transformations, such as cos(θ - 180°), is essential for determining its sign.

Angle Transformation

Angle transformation involves adjusting an angle by adding or subtracting a specific value. In this case, cos(θ - 180°) indicates a shift of the angle θ by 180°, which effectively reflects the angle across the origin on the unit circle. This transformation changes the sign of the cosine value, making it important to analyze how this affects the function's output based on the original angle θ.