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Chapter 5: Problem 37

For the following exercises, convert angles in degrees to radians. $$ 180^{\circ} $$

Short Answer

Expert verified
180掳 is equal to 蟺 radians.

Step by step solution

01

Understand the Conversion Formula

To convert degrees to radians, use the formula \( ext{Radians} = ext{Degrees} imes \frac{\pi}{180} \). This formula takes an angle in degrees and converts it to an angle in radians.
02

Substitute the Given Angle into the Formula

Substitute the given angle, \(180^{\circ}\), into the conversion formula. You will have \(180^{\circ} \times \frac{\pi}{180}\).
03

Simplify the Expression

Simplify the expression by cancelling out the \(180\) from the numerator and denominator. This leaves you with \(\pi\).
04

Final Result

After simplifying, the angle \(180^{\circ}\) converts to \(\pi\) radians.

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Key Concepts

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

Understanding Degrees to Radians
Angles can be measured in either degrees or radians. These are two different systems for describing an angle. Understanding this becomes critical when you're dealing with trigonometric functions and calculus.
Degrees is a familiar unit, with a complete circle measuring 360 degrees. However, radians offer a different perspective, treating the angle in context with the circle's arc.
This is pivotal in many branches of mathematics and physics, where radian measure is often more convenient than degrees.
The Conversion Formula
To convert degrees into radians, we rely on the conversion formula:
  • \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]
This formula emerges from the relationship of the full circle in degrees (360) and radians (\(2\pi\)).
Understanding this formula is key to smoothly transition between these units, maintaining consistency in mathematical equations and real-world applications.
This formula encapsulates the bridge from the degree system into the radian system.
Simplification Steps
Simplifying a conversion from degrees to radians involves a series of straightforward mathematical steps.
Let's go through the simplification process using the angle of 180 degrees:
  • Substitute 180 degrees into the conversion formula: \(180 \times \frac{\pi}{180}\).
  • The "180" in both the numerator and the denominator simplifies, effectively cancelling each other out.
  • You are then left with \(\pi\), signifying the radian measure of 180 degrees.
These steps are crucial in streamlining the conversion process and ensuring accuracy.
Interpreting the Radian Measure
Once we have simplified and calculated the radian measure, it is essential to understand what this new unit represents.
In our example, the conversion of 180 degrees to radians results in \(\pi\) radians.
This indicates that the angle forms a semicircle on the unit circle.
This conversion is not just about changing numbers; it offers a deeper understanding of the angular relationships as they relate to the circle's circumference. Thus, using radians can often simplify mathematical problems, especially those involving period functions or circular motion.

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Most popular questions from this chapter

For the following exercises, draw an angle in standard position with the given measure. $$ -\frac{\pi}{6} $$

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