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Chapter 5: Problem 58

Convert each degree measure to radian measure. Give exact answers. $$-105^{\circ}$$

Short Answer

Expert verified
-\( -105^{\circ} \) is \(-\frac{7\pi}{12}\) radians.

Step by step solution

01

Recall the Conversion Factor

To convert degrees to radians, use the conversion factor: \( \frac{\pi}{180^{\circ}} \).
02

Set Up the Conversion

Multiply the given degree measure by the conversion factor. In this case: \( -105^{\circ} \times \frac{\pi}{180^{\circ}} \).
03

Simplify the Expression

Simplify the multiplication: \( -105 \times \frac{\pi}{180} = -\frac{105\pi}{180} \).
04

Reduce the Fraction

Reduce the fraction by finding the greatest common divisor of 105 and 180. Both can be divided by 15: \( -\frac{105\pi}{180} = -\frac{105 \div 15 \pi}{180 \div 15} = -\frac{7\pi}{12} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

degree to radian conversion
Understanding how to convert degrees to radians is essential in trigonometry. The process involves using a conversion factor, which is \( \frac{\pi}{180^{\circ}} \). This factor comes from the relationship between degrees and radians, where a full circle measures 360 degrees or \( 2\pi \) radians. To convert an angle in degrees to radians, multiply the degree measure by this factor.
angle measurement
Angles can be measured in degrees or radians. Degrees are more common in daily life, like in navigation and construction. Radians, however, are more useful in higher mathematics and physics. One radian is the angle made when the radius is wrapped along the circle's circumference. Since there are \( 2\pi \) radians in a full circle, we can derive that \( 180^{\circ} = \pi \) radians. This relationship is the basis for converting between the two units.
trigonometry
Trigonometry is the study of the relationships between the angles and sides of triangles. It is fundamental in many areas of science and engineering. Concepts like sine, cosine, and tangent functions are based on angle measurement. Understanding how to convert between degrees and radians is crucial because many trigonometric formulas are derived using radians. For instance, the unit circle, a key concept in trigonometry, is defined using radians.

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Most popular questions from this chapter

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=2 \sec x$$

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