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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2:48 minutes

Problem 49

Textbook Question

Use a calculator to evaluate each expression.cos 75°29' cos 14°31' - sin 75°29' sin 14°31'

Verified step by step guidance

Recognize that the expression cos 75°29' cos 14°31' - sin 75°29' sin 14°31' is in the form of the cosine of a sum identity: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).

Identify the angles: \( A = 75°29' \) and \( B = 14°31' \).

Apply the cosine of a sum identity: \( \cos(75°29' + 14°31') \).

Add the angles: \( 75°29' + 14°31' = 90° \).

Conclude that the expression simplifies to \( \cos(90°) \).

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Key Concepts

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

Cosine and Sine Functions

Cosine and sine are fundamental trigonometric functions that relate the angles of a triangle to the ratios of its sides. The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse, while the sine is the ratio of the opposite side to the hypotenuse. These functions are periodic and play a crucial role in various applications, including wave motion and oscillations.

Angle Addition Formula

The angle addition formulas for sine and cosine allow us to express the sine or cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, cos(A + B) = cosA cosB - sinA sinB. This formula is essential for simplifying expressions involving trigonometric functions of combined angles, as seen in the given expression.

Degrees and Minutes

In trigonometry, angles can be measured in degrees and minutes, where one degree is divided into 60 minutes. This notation is particularly useful for expressing angles that are not whole numbers. To evaluate trigonometric functions for angles given in degrees and minutes, it is often necessary to convert them into decimal degrees or use a calculator that can handle this format directly.