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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1:32 minutes

Problem 33

Textbook Question

Find the measure of the smaller angle formed by the hands of a clock at the following times.

Verified step by step guidance

Step 1: Understand the problem. We need to find the smaller angle between the hour and minute hands of a clock at a given time.

Step 2: Use the formula for the angle between the hour and minute hands. The angle \( \theta \) in degrees can be calculated using \( \theta = |30h - 5.5m| \), where \( h \) is the hour and \( m \) is the minutes.

Step 3: Substitute the given time into the formula. Identify the hour \( h \) and the minutes \( m \) from the given time.

Step 4: Calculate the absolute value of the expression \( |30h - 5.5m| \) to find the angle.

Step 5: Determine if the calculated angle is the smaller angle. If the angle is greater than 180 degrees, subtract it from 360 degrees to find the smaller angle.

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Key Concepts

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

Clock Angle Formula

The clock angle formula calculates the angle between the hour and minute hands of a clock. It is given by the formula: Angle = |(30*hour - (11/2)*minutes)|. This formula accounts for the fact that the hour hand moves as the minutes pass, providing an accurate measure of the angle formed at any given time.

Understanding Clock Positions

To effectively use the clock angle formula, one must understand the positions of the hour and minute hands. The hour hand moves 30 degrees for each hour (360 degrees/12 hours), while the minute hand moves 6 degrees for each minute (360 degrees/60 minutes). Recognizing these movements is essential for determining the correct angle.

Finding the Smaller Angle

When calculating the angle between the clock hands, it is important to find the smaller angle, which is less than or equal to 180 degrees. If the calculated angle exceeds 180 degrees, the smaller angle can be found by subtracting it from 360 degrees. This ensures that the answer reflects the actual angle formed between the hands.