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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3:03 minutes

Problem 93

Textbook Question

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

Verified step by step guidance

Recognize that \( \theta \) is in the second quadrant since \( 90^\circ < \theta < 180^\circ \).

Recall that the cotangent function, \( \cot(\theta) \), is negative in the second quadrant because \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and \( \cos(\theta) \) is negative while \( \sin(\theta) \) is positive in this quadrant.

Use the identity \( \cot(\theta + 180^\circ) = \cot(\theta) \) because adding 180° to an angle results in the same cotangent value but in the opposite direction on the unit circle.

Since \( \cot(\theta) \) is negative in the second quadrant, \( \cot(\theta + 180^\circ) \) will have the same sign as \( \cot(\theta) \) in the fourth quadrant, where cotangent is positive.

Conclude that \( \cot(\theta + 180^\circ) \) is positive.

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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 in different quadrants is essential, as its sign depends on the signs of sine and cosine in those quadrants.

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 determining the sign of cotangent in the specified range of θ.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each with distinct signs for sine and cosine. In the second quadrant (90° < θ < 180°), sine is positive and cosine is negative. This understanding is vital for determining the sign of cotangent, as it relies on the signs of sine and cosine.