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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2:44 minutes

Problem 48

Textbook Question

Use a calculator to evaluate each expression.2 sin 25°13' cos 25°13' - sin 50°26'

Verified step by step guidance

Recognize that the expression can be simplified using trigonometric identities.

Recall the double angle identity: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \).

Notice that \( 2 \sin(25°13') \cos(25°13') \) can be rewritten using the double angle identity as \( \sin(2 \times 25°13') \).

Calculate \( 2 \times 25°13' \) to find the angle for the sine function, which is \( 50°26' \).

Substitute back into the expression: \( \sin(50°26') - \sin(50°26') \), which simplifies to zero.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), relate the angles of a triangle to the ratios of its sides. In this context, sin and cos are used to evaluate the expressions involving angles measured in degrees and minutes. Understanding how these functions operate is essential for simplifying and calculating trigonometric expressions.

Angle Conversion

In trigonometry, angles can be expressed in degrees, minutes, and seconds. The notation 25°13' indicates 25 degrees and 13 minutes. It is crucial to convert these angles into a decimal format or to understand their relationship in calculations, especially when using a calculator that may require angles in decimal degrees.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One relevant identity is sin(2θ) = 2sin(θ)cos(θ), which can simplify expressions like the one in the question. Recognizing and applying these identities can facilitate the evaluation of complex trigonometric expressions.