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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Problem 2

Textbook Question

CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. Given tan θ = 1/cot θ , two equivalent forms of this identity are cot θ = 1/ and tan θ . = 1 .

Verified step by step guidance

Recognize that $ \tan \theta $ and $ \cot \theta $ are reciprocal trigonometric functions, meaning $ \tan \theta = \frac{1}{\cot \theta} $ and $ \cot \theta = \frac{1}{\tan \theta} $.

To fill in the first blank, use the reciprocal relationship: $ \cot \theta = 1/\tan \theta $.

For the second blank, use the identity $ \tan \theta \cdot \cot \theta = 1 $, which follows from the reciprocal nature of $ \tan \theta $ and $ \cot \theta $.

Thus, the second blank is filled with $ \cot \theta $, making the equation $ \tan \theta \cdot \cot \theta = 1 $.

These identities are useful for simplifying expressions and solving trigonometric equations.

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Key Concepts

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

Reciprocal Identities

Reciprocal identities in trigonometry express the relationship between the primary trigonometric functions and their reciprocals. For example, the tangent function (tan θ) is the reciprocal of the cotangent function (cot θ), meaning tan θ = 1/cot θ. Understanding these identities is crucial for manipulating and solving trigonometric equations.

Trigonometric Functions

Trigonometric functions, including sine, cosine, tangent, cotangent, secant, and cosecant, are fundamental in trigonometry. Each function relates an angle of a right triangle to the ratios of its sides. Recognizing how these functions interrelate, such as tan θ and cot θ, is essential for solving problems involving angles and their measures.

Identity Verification

Identity verification in trigonometry involves proving that two expressions are equivalent for all values of the variable within a certain domain. This is often done by manipulating one side of the equation to match the other using known identities. Mastery of this concept allows students to confidently fill in blanks or complete identities in trigonometric equations.

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What is a positive value of A in the interval $\left[0\degree,90\degree\right)$ that will make the following statement true? Express the answer in four decimal places.

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