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

Trigonometric Functions on the Unit Circle

Trigonometry

3. Unit Circle

Trigonometric Functions on the Unit Circle

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2:09 minutes

Problem 9

Textbook Question

Determine whether each statement is true or false. If false, tell why.tan 60° ≥ cot 40°

Verified step by step guidance

Convert the given angles to radians if necessary, but since they are in degrees, we can work with them directly.

Recall the trigonometric identities: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} \).

Calculate \( \tan(60^\circ) \) using the identity \( \tan(60^\circ) = \sqrt{3} \).

Calculate \( \cot(40^\circ) \) using the identity \( \cot(40^\circ) = \frac{1}{\tan(40^\circ)} \).

Compare \( \tan(60^\circ) \) and \( \cot(40^\circ) \) to determine if \( \tan(60^\circ) \geq \cot(40^\circ) \) is true or false.

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Key Concepts

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

Tangent and Cotangent Functions

The tangent function, denoted as tan(θ), is the ratio of the opposite side to the adjacent side in a right triangle. The cotangent function, cot(θ), is the reciprocal of the tangent, defined as cot(θ) = 1/tan(θ). Understanding these functions is essential for comparing their values, especially at specific angles like 60° and 40°.

Angle Relationships in Trigonometry

In trigonometry, certain angles have known values for their tangent and cotangent. For example, tan(60°) equals √3, while cot(40°) can be calculated as 1/tan(40°). Recognizing these relationships allows for accurate comparisons between different trigonometric functions at specified angles.

Inequalities in Trigonometric Functions

When comparing trigonometric values, understanding inequalities is crucial. The statement tan(60°) ≥ cot(40°) requires evaluating both sides to determine if the inequality holds true. This involves calculating or estimating the values of tan(60°) and cot(40°) to draw a valid conclusion about the relationship between them.