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Chapter 8: Problem 4

Convert the following to radian measure. $$450^{\circ},-210^{\circ},-90^{\circ}$$

Short Answer

Expert verified
450掳 = \(\frac{5\pi}{2}\) radians, -210掳 = \(-\frac{7\pi}{6}\) radians, and -90掳 = \(-\frac{\pi}{2}\) radians.

Step by step solution

01

- Understand the Conversion Factor

To convert degrees to radians, use the conversion factor where \( 180^{\circ} = \pi \text{ radians} \). This implies \( 1^{\circ} = \frac{\pi}{180} \text{ radians} \).
02

- Convert 450 Degrees to Radians

Multiply 450 degrees by \( \frac{\pi}{180} \) to convert to radians: \(450^{\circ} \times \frac{\pi}{180} = \frac{450\pi}{180} = 2\pi \). So, 450 degrees is \(\frac{5\pi}{2} \) radians.
03

- Convert -210 Degrees to Radians

Similar to Step 2, multiply -210 degrees by \( \frac{\pi}{180} \): \(-210^{\circ} \times \frac{\pi}{180} = \frac{-210\pi}{180} = -\frac{7\pi}{6} \). Thus, -210 degrees is \(-\frac{7\pi}{6} \) radians.
04

- Convert -90 Degrees to Radians

Finally, convert -90 degrees by multiplying by \(\frac{\pi}{180}\): \(-90^{\circ} \times \frac{\pi}{180} = \frac{-90\pi}{180} = -\frac{\pi}{2} \). Therefore, -90 degrees is \(-\frac{\pi}{2} \) radians.

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Key Concepts

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

Radian Measure
Radian measure is a way of expressing angles using the radius of a circle. Instead of relying on degrees, radians relate directly to the length of the arc that an angle subtends on a unit circle. A unit circle has a radius of one unit, and a full circle is \(\theta = 2\pi\) radians.
This means:
  • A quarter circle, or 90 degrees, is \(\theta = \frac{\pi}{2}\) radians
  • A half circle, or 180 degrees, is \(\theta = \pi\) radians
  • Three-quarters of a circle, or 270 degrees, is \(\theta = \frac{3\pi}{2}\) radians
This measure is particularly useful in calculus and trigonometry as it simplifies the formulation of many mathematical expressions. Radians make it easier to understand the relationship between the arc length and the radius.
Degrees
Degrees are the more familiar unit for measuring angles, often used in everyday activities. They divide a circle into 360 equal parts, where each part is one degree. This subdivision is handy for practical geometry and navigation. Here are some key points about degrees:
  • One full circle is 360 degrees
  • Right angles are 90 degrees
  • Straight angles are 180 degrees
Degrees offer an intuitive understanding of angles and are commonly used in various fields, including construction, navigation, and education. However, switching between degrees and radians is essential when working with more advanced mathematics.
Conversion Factor
To convert between degrees and radians, we use the conversion factor that ties these two measurement systems together. The relationship is that 180 degrees equals \(\theta = \pi\) radians.
Therefore, we derive:
  • \

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Most popular questions from this chapter

Differentiate (with respect to \(t\) or \(x\) ): $$f(t)=\tan \pi t$$

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Find \(t\) such that \(0 \leq t \leq \pi\) and \(t\) satisfies the stated condition. $$\cos t=\cos (-\pi / 6)$$

Construct angles with the following radian measure. $$\pi / 3,5 \pi / 2,6 \pi$$

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