Textbook Question
Work each problem.
Consider each angle in standard position having the given radian measure. In what quadrant does the terminal side lie?
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1. Measuring Angles
Complementary and Supplementary Angles
Problem 3.5
Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π .
45°
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: \(45^\circ \times \frac{\pi}{180}\).
Simplify the fraction \(\frac{45}{180}\) by dividing numerator and denominator by their greatest common divisor.
Express the simplified fraction multiplied by \(\pi\) to write the answer as a multiple of \(\pi\).
Write the final answer in the form \(\frac{\text{numerator}}{\text{denominator}}\pi\) without calculating the decimal value.
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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 mathematical contexts, especially calculus and trigonometry.
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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 result exact and avoids decimal approximations, which is preferred in trigonometry.
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Simplifying Fractions in Radian Measures
After converting degrees to radians, the resulting fraction involving π should be simplified if possible. Simplifying fractions makes the radian measure clearer and easier to use in further calculations or interpretations.
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