Problem 29 - Precalculus with Calculus Previews

Question

In Problems \(25-32,\) convert the given angle from degrees to radians. $$ 270^{\circ} $$

Step-by-Step Solution

Answer

The angle \(270^{\circ}\) converts to \(\frac{3\pi}{2}\) radians.

Step 1 - Understand the Conversion Formula

To convert degrees to radians, we use the formula: \[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \] This formula arises because a full circle is \(360^{\circ}\), which is equal to \(2\pi\) radians.

Step 2 - Substitute the Given Angle into the Formula

Substitute the given angle, \(270^{\circ}\), into the degrees-to-radians conversion formula:\[ 270^{\circ} \times \frac{\pi}{180} \]

Step 3 - Simplify the Expression

Simplify the fraction:\[ 270 \times \frac{\pi}{180} = \frac{270\pi}{180} \]Then, simplify \(\frac{270}{180}\) by dividing both the numerator and denominator by their greatest common divisor (90):\[ \frac{270}{180} = \frac{3}{2} \]Thus, the expression becomes:\[ \frac{3\pi}{2} \]

Step 4 - Conclude the Conversion

Therefore, the angle \(270^{\circ}\) is equivalent to \(\frac{3\pi}{2}\) radians.

Key Concepts

Angle Measurement

Angles can be measured in different units, primarily degrees and radians. These units help us describe the size of an angle, showing how much we have rotated around a point. When you picture an angle, imagine two lines (or rays) coming together at a point known as the vertex. The amount of turn between these lines is what we call the angle.
In everyday activities, such as navigation and weather patterns, angles are usually measured in degrees. A full circle is divided into 360 equal parts, which means a full circle represents 360 degrees (°). A quarter of a circle, often referred to as a right angle, is 90 degrees.
The degree system is familiar and extensively used because it divides the circle nicely for everyday purposes like geometry and navigation. However, when working in mathematics and science, radians are often preferable.

Conversion Formula

To convert an angle from degrees to radians, we utilize a straightforward formula: \[ \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \] This conversion formula takes advantage of the fact that a full circle can be represented in both degrees and radians.
A circle is 360 degrees, while in radians, a full circle is represented by \(2\pi\). Therefore, the conversion factor \(\frac{\pi}{180}\) arises by equating the angular measures in degrees and radians. It tells us how much a single degree is worth in radians.
Here's the step-by-step approach:

• Take the degree measure you have. For example, in this exercise, it's 270°.
• Multiply it by \(\frac{\pi}{180}\). Why \(\frac{\pi}{180}\)? Because \(180^{\circ}\) equals \(\pi\) radians.
• Simplify the result where possible, by dividing the numbers appropriately to reach the simplified radian measure.

Radian

In many scientific and engineering contexts, radians provide a more natural approach to measuring angles. But what exactly is a radian?
A radian is the angle created when the arc length is equal to the radius of the circle. Essentially, it binds the concept of linear distance (the arc) with angular measurement. If a circle's circumference is \(2\pi\) times its radius, then the equivalent of 360 degrees in radians is \(2\pi\).
Using radians can simplify many mathematical equations and formulas, particularly calculus, as it relates naturally to the properties of circles. For example:

• A right angle, or 90 degrees, equals \(\frac{\pi}{2}\) radians as it occupies a quarter of the circle.
• Similarly, our exercise with 270 degrees becomes \(\frac{3\pi}{2}\) radians, indicating three-quarters of a circle.

Understanding radians is crucial for those delving into advanced math or science, as they offer a clean and concise way to handle circular and periodic phenomena.