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Chapter 6: Problem 15

Convert to radian measure. Leave the answer in terms of \(\pi\). $$-180^{\circ}$$

Short Answer

Expert verified
-\pi radians

Step by step solution

01

Understand the relationship between degrees and radians

Remember that to convert degrees to radians, the conversion factor is \( \frac{\pi}{180} \). This is because \( 180^{\circ} = \pi \) radians.
02

Set up the conversion

Multiply the given angle in degrees by \( \frac{\pi}{180} \). In this case, it is \( -180^{\circ} \). So, we have: \[ -180^{\circ} \times \frac{\pi}{180} \]
03

Simplify the expression

Simplify the multiplication: \[ -180 \times \frac{\pi}{180} = -\pi \] Therefore, the radian measure is \( -\pi \).

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Key Concepts

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

degrees to radians
To convert degrees to radians, we need to understand the relationship between these two units of measurement. Degrees and radians are both ways to measure angles. Degrees are more commonly used in everyday situations, while radians are more prevalent in higher mathematics, especially in trigonometry and calculus.

The key relationship to remember is that a full circle is 360 degrees, and the same full circle is \( 2\pi \) radians. Hence, half a circle, or 180 degrees, is equal to \( \pi \) radians. This core relationship forms the basis for conversions between degrees and radians.
conversion factor
The conversion factor is the bridge that allows us to translate an angle measurement from degrees to radians. Given the relationship where 180 degrees is equal to \( \pi \) radians, our conversion factor is \( \frac{\pi}{180} \). This factor tells us how many radians are in one degree.

To convert degrees to radians, you simply multiply the number of degrees by this conversion factor. For instance, to convert \( -180^{\circ} \) to radians, we use the formula:
  • \( -180^{\circ} \times \frac{\pi}{180} = -\pi \)
This formula simplifies our degrees directly into radians using the conversion factor.
simplifying expressions
Once we have set up our conversion using the conversion factor, we often need to simplify the expression to get our final result. Simplifying the multiplication in our example is straightforward:
  • \( -180^{\circ} \times \frac{\pi}{180} \)
We can see the 180 degrees in the numerator and denominator cancel each other out:
  • \( -\frac{180\pi}{180} \)
Which simplifies to:
  • \( -\pi \)

This leaves us with the final, simplified radian measure: \( -\pi \). Simplifying expressions like this is crucial for getting a clean, understandable result.

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