Textbook Question
Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5.
r = 12.7 cm, θ = 81°
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Ch. 3 - Radian Measure and The Unit Circle
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 4, Problem 54
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
122° 37'
Verified step by step guidance1
Understand that to convert degrees to radians, we use the formula: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
First, convert the given angle from degrees and minutes to a decimal degree. Recall that 1 minute (') is equal to \(\frac{1}{60}\) of a degree, so convert 37' to degrees by calculating \(37 \times \frac{1}{60}\).
Add the decimal degree value from the minutes to the whole degrees: \(122 + \left(37 \times \frac{1}{60}\right)\) to get the total degrees in decimal form.
Multiply the decimal degree value by \(\frac{\pi}{180}\) to convert it to radians: \(\left(122 + \frac{37}{60}\right) \times \frac{\pi}{180}\).
If needed, use a calculator to approximate the radian value and round it to the nearest thousandth.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Degree to Radian Conversion
Degrees and radians are two units for measuring angles. To convert degrees to radians, multiply the degree measure by π/180. This conversion is essential because radians are the standard unit in many trigonometric applications.
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Converting between Degrees & Radians
Converting Minutes to Decimal Degrees
Angle measurements often include minutes (') where 1 degree equals 60 minutes. To convert minutes to decimal degrees, divide the minutes by 60 and add this to the degree value. This step is necessary before converting the total degrees to radians.
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5:04Converting between Degrees & Radians
Rounding to the Nearest Thousandth
After converting to radians, the result may be an irrational number. Rounding to the nearest thousandth means keeping three decimal places, which balances precision and simplicity for practical use in calculations.
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Related Practice
Textbook Question
Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. See Example 5.
r = 40.0 mi, θ = 135°
Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
-47.69°
Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
56° 25'
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Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
cos 2
Textbook Question
Convert each degree measure to radians. If applicable, round to the nearest thousandth. See Example 1(c).
85.04°
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