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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2:35 minutes

Problem 10

Textbook Question

Determine whether each statement is true or false. If false, tell why.csc 22° ≤ csc 68°

Verified step by step guidance

Step 1: Recall that the cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function, \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Step 2: Understand that the sine function, \( \sin(\theta) \), is increasing in the interval \( 0^\circ \) to \( 90^\circ \). Therefore, \( \sin(22^\circ) < \sin(68^\circ) \).

Step 3: Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), the cosecant function is decreasing in the interval \( 0^\circ \) to \( 90^\circ \).

Step 4: Given that \( \sin(22^\circ) < \sin(68^\circ) \), it follows that \( \csc(22^\circ) > \csc(68^\circ) \).

Step 5: Conclude that the statement \( \csc(22^\circ) \leq \csc(68^\circ) \) is false because \( \csc(22^\circ) > \csc(68^\circ) \).

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Key Concepts

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

Cosecant Function

The cosecant function, denoted as csc, is the reciprocal of the sine function. For any angle θ, csc(θ) = 1/sin(θ). This means that the value of csc is determined by the sine of the angle, and it is undefined when sin(θ) = 0. Understanding the behavior of the sine function is crucial for evaluating the cosecant values.

Angle Relationships in Trigonometry

In trigonometry, the angles 22° and 68° are complementary, meaning they add up to 90°. This relationship implies that sin(22°) = cos(68°), which can be used to compare their cosecant values. Recognizing complementary angles helps in understanding how trigonometric functions relate to one another.

Inequalities in Trigonometric Functions

When comparing trigonometric functions, it is essential to understand their ranges and behaviors. Since csc(θ) decreases as θ increases in the first quadrant (0° to 90°), we can infer that csc(22°) will be greater than csc(68°). This concept is vital for determining the truth of the given inequality.