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Chapter 5: Problem 34

For the following exercises, convert angles in degrees to radians. $$ 100^{\circ} $$

Short Answer

Expert verified
100掳 is \( \frac{5\pi}{9} \) radians.

Step by step solution

01

Understand the Relationship Between Degrees and Radians

To convert an angle from degrees to radians, we use the relationship that \( 180^{\circ} \) is equal to \( \pi \) radians. Therefore, the conversion factor is \( \frac{\pi}{180} \).
02

Apply the Conversion Formula

To convert \( 100^{\circ} \) to radians, multiply \( 100 \) by \( \frac{\pi}{180} \). The formula is: \[ 100^{\circ} \times \frac{\pi}{180} = \frac{100\pi}{180} \]
03

Simplify the Expression

Now simplify the fraction \( \frac{100\pi}{180} \). Divide both the numerator and the denominator by their greatest common divisor, which is 20.\[ \frac{100\pi}{180} = \frac{5\pi}{9} \]
04

Conclusion with the Answer

The final result, which is \( 100^{\circ} \) converted to radians, is \( \frac{5\pi}{9} \).

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Key Concepts

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

Degrees to Radians
When we talk about angles, we can express them in two main units: degrees and radians. Understanding how to switch between these units is crucial, especially in higher mathematics like calculus. So, what's the difference?
Degrees measure angles by dividing a circle into 360 parts, while radians use the circle's radius as the unit of measure. A full circle has an angle of \( 2\pi \) radians or \( 360^{\circ} \). This means \( 180^{\circ} \) is equal to \( \pi \) radians.
Using this relationship helps us convert any angle from degrees to radians with ease. Ready to learn the magic formula that makes this conversion simple?
Conversion Formula
The magic behind converting degrees to radians lies in a simple formula. The process leverages the relationship that \( 180^{\circ} = \pi \) radians. Hence, the conversion factor is \( \frac{\pi}{180} \).
Here's a step-by-step way to use it:
  • First, take the angle you have in degrees. In our example, it's \( 100^{\circ} \).
  • Multiply the degree measure by the conversion factor: \( \frac{\pi}{180} \).
  • The formula becomes: \( 100^{\circ} \times \frac{\pi}{180} \).
This formula converts the degree measure into radians by scaling it from the 360-degree circle format to the \( 2\pi \) radian format. Coming up next, we'll see how to simplify this multiplication.
Simplifying Fractions
Once you've multiplied an angle by the conversion factor, you'll often find yourself with a fraction. Simplifying this fraction makes it easier to use in calculations.
Let's look at our example: after applying the conversion formula, we have \( \frac{100\pi}{180} \). To simplify:
  • Find the greatest common divisor (GCD) of the numerator and the denominator. For \( 100 \) and \( 180 \), it's \( 20 \).
  • Divide both the numerator and the denominator by the GCD.
  • This results in \( \frac{5\pi}{9} \) after simplifying \( \frac{100\pi}{180} \) by dividing both parts by \( 20 \).
By simplifying fractions, you ensure that your answers are in their most usable and comprehensible forms, making your math work more accurate and efficient.

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Most popular questions from this chapter

The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

For the following exercises, draw an angle in standard position with the given measure. $$ -150^{\circ} $$

For the following exercises, draw the angle provided in standard position on the Cartesian plane. $$ -\frac{\pi}{3} $$

For the following exercises, find the exact value of the given expression. $$ \sec \frac{\pi}{4} $$

For the following exercises, use reference angles to evaluate the given expression. If \(\sec (t)=-2.5,\) what is the \(\sec (-t) ?\)
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