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

Previous problemNext problem

Problem 80

Textbook Question

Suppose θ is in the interval (90°, 180°). Find the sign of each of the following.cot(θ + 180°)

Verified step by step guidance

Recognize that \( \theta \) is in the second quadrant, where angles range from 90° to 180°.

Recall that \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). In the second quadrant, \( \cos(\theta) < 0 \) and \( \sin(\theta) > 0 \), so \( \cot(\theta) < 0 \).

Use the identity \( \cot(\theta + 180°) = \cot(\theta) \).

Since \( \cot(\theta) < 0 \) in the second quadrant, \( \cot(\theta + 180°) \) will have the same sign as \( \cot(\theta) \).

Conclude that \( \cot(\theta + 180°) < 0 \).

Verified video answer for a similar problem:

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

Play a video:

Was this helpful?

Key Concepts

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

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function. It is defined as cot(θ) = cos(θ) / sin(θ). Understanding the behavior of cotangent is essential for determining its sign in various quadrants of the unit circle.

Angle Addition

The angle addition formula allows us to find the trigonometric function of a sum of angles. For cotangent, cot(θ + 180°) can be simplified using the identity cot(θ + 180°) = cot(θ). This property is crucial for evaluating the sign of cotangent in different intervals.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle measures. In the second quadrant (90° to 180°), sine is positive and cosine is negative, which affects the sign of cotangent. Recognizing the quadrant in which θ lies helps determine the sign of cot(θ + 180°).