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°
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1. Measuring Angles
Radians
Problem 11
Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π. See Examples 1(a) and 1(b). 60°
Verified step by step guidance1
Recall the formula to convert degrees to radians: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).
Substitute the given degree measure into the formula: \(60^\circ \times \frac{\pi}{180}\).
Simplify the fraction \(\frac{60}{180}\) by dividing numerator and denominator by their greatest common divisor, which is 60.
After simplification, express the result as a multiple of \(\pi\).
Write the final answer in radians as a simplified fraction times \(\pi\).
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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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Understanding π as a Constant
π (pi) is an irrational constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter. Expressing answers as multiples of π keeps the results exact and simplifies further calculations in trigonometry.
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Simplifying Fractions
After converting degrees to radians, the resulting fraction involving π should be simplified to its lowest terms. Simplification makes the expression clearer and easier to interpret, such as converting 60° to π/3 radians.
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