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

1. Measuring Angles

Angles in Standard Position

Trigonometry

1. Measuring Angles

Angles in Standard Position

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2:42 minutes

Problem 27

Textbook Question

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1. 1—————— csc(90°-51°)

Verified step by step guidance

Recognize that \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Use the co-function identity: \( \csc(90^\circ - \theta) = \sec(\theta) \).

Substitute \( \theta = 51^\circ \) into the identity: \( \csc(90^\circ - 51^\circ) = \sec(51^\circ) \).

Simplify the expression: \( \frac{1}{\csc(90^\circ - 51^\circ)} = \frac{1}{\sec(51^\circ)} \).

Use a calculator to find \( \cos(51^\circ) \) and approximate the value of \( \frac{1}{\sec(51^\circ)} = \cos(51^\circ) \) to six decimal places.

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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. It is defined as csc(θ) = 1/sin(θ). Understanding this relationship is crucial for simplifying expressions involving cosecant, especially when working with angles in trigonometric identities.

Co-Function Identity

Co-function identities state that the sine of an angle is equal to the cosine of its complement. Specifically, sin(90° - θ) = cos(θ). This identity is essential for simplifying expressions like csc(90° - 51°) by transforming it into a more manageable form.

Calculator Usage for Trigonometric Functions

Using a calculator to evaluate trigonometric functions requires understanding the angle measurement mode (degrees or radians). For this problem, ensure the calculator is set to degrees to accurately compute values like sin(51°) and subsequently find csc(90° - 51°) to six decimal places.