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

Defining the Unit Circle

Trigonometry

3. Unit Circle

Defining the Unit Circle

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Problem 3.39

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3. csc (―9.4946)

Verified step by step guidance

Recognize that the cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function, \( \sin(\theta) \). Therefore, \( \csc(-9.4946) = \frac{1}{\sin(-9.4946)} \).

Use the property of the sine function that \( \sin(-\theta) = -\sin(\theta) \) to simplify the expression: \( \csc(-9.4946) = \frac{1}{-\sin(9.4946)} \).

Calculate \( \sin(9.4946) \) using a calculator to find its approximate value.

Take the reciprocal of the calculated sine value to find \( \csc(9.4946) \).

Round the result to four decimal places to obtain the final approximation for \( \csc(-9.4946) \).

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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 to find the cosecant of an angle, one must first determine the sine of that angle and then take its reciprocal. Understanding this relationship is crucial for calculating csc values accurately.

Calculator Functions

Using a scientific calculator effectively is essential for approximating trigonometric functions. Most calculators have dedicated buttons for sine, cosine, and tangent, as well as their reciprocals. Familiarity with how to input angles in radians or degrees, depending on the calculator's settings, is necessary to obtain correct results for functions like csc.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius. In trigonometry, many functions are often evaluated in radians, so converting degrees to radians or understanding when to use radians is vital for accurate calculations, especially when dealing with non-standard angles.