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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1:56 minutes

Problem 45

Textbook Question

Determine whether each statement is possible or impossible. a. sec θ = ―2/3

Verified step by step guidance

Recall that the secant function, \( \sec \theta \), is the reciprocal of the cosine function, \( \cos \theta \). Therefore, \( \sec \theta = \frac{1}{\cos \theta} \).

Given \( \sec \theta = -\frac{2}{3} \), this implies \( \cos \theta = -\frac{3}{2} \).

The range of the cosine function is \([-1, 1]\), meaning \( \cos \theta \) can only take values within this interval.

Since \(-\frac{3}{2}\) is outside the range of \([-1, 1]\), it is impossible for \( \cos \theta \) to equal \(-\frac{3}{2}\).

Therefore, the statement \( \sec \theta = -\frac{2}{3} \) is impossible.

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Key Concepts

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

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). Since the cosine function can only take values between -1 and 1, the secant function will have values outside this range, specifically less than -1 or greater than 1.

Range of the Secant Function

The range of the secant function is important for determining the validity of sec(θ) values. Specifically, sec(θ) can take any value less than -1 or greater than 1. Therefore, a value like -2/3, which lies between -1 and 1, is not possible for sec(θ).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Understanding these identities helps in analyzing the relationships between different trigonometric functions, such as how sec(θ) relates to cos(θ). This knowledge is crucial for determining the feasibility of given trigonometric statements.