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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1:17 minutes

Problem 25

Textbook Question

Concept Check What is wrong with the following item that appears on a trigonometry test? "Find sec θ , given that cos θ = 3/2 . "

Verified step by step guidance

Step 1: Recall the definition of secant in terms of cosine. The secant function is the reciprocal of the cosine function, so \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Substitute the given value of \( \cos \theta = \frac{3}{2} \) into the reciprocal formula for secant.

Step 3: Recognize that the cosine of an angle cannot be greater than 1, as the range of the cosine function is \([-1, 1]\).

Step 4: Identify the error in the problem: \( \cos \theta = \frac{3}{2} \) is not possible for any real angle \( \theta \).

Step 5: Conclude that the problem is incorrect because it provides an impossible value for \( \cos \theta \).

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Key Concepts

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

Definition of Secant

Secant is defined as the reciprocal of the cosine function. Mathematically, sec(θ) = 1/cos(θ). Understanding this relationship is crucial for solving problems involving secant, as it directly relates to the value of cosine.

Range of the Cosine Function

The cosine function has a range of values between -1 and 1, meaning that cos(θ) can never exceed 1 or be less than -1. Therefore, a value like 3/2 is invalid for cos(θ), indicating a fundamental misunderstanding of the function's properties.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Recognizing that sec(θ) cannot be calculated from an invalid cosine value requires familiarity with these identities and their constraints.