Problem 36 - Contemporary Precalculus

Question

Convert the given degree measure to radians. $$252^{\circ}$$

Step-by-Step Solution

Answer

Question: Convert the degree measure 252° to radians. Answer: $$\frac{7\pi}{5} \, radians$$

Step 1: Write down the given degree measure and the conversion factor

We are given the degree measure as $$252^{\circ}$$ and the conversion factor is $$1^{\circ} = \frac{\pi}{180} \, radians$$.

Step 2: Multiply the given degree measure by the conversion factor

To find the equivalent measure in radians, we will multiply the given degree measure by the conversion factor: $$(252^{\circ})\times \frac{\pi}{180} \, radians$$

Step 3: Simplify the expression

Now, we will simplify the expression to find the equivalent measure in radians: $$\frac{252\pi}{180} \, radians$$

Step 4: Reduce the fraction to the lowest terms

To reduce the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 36: $$\frac{252\div 36\pi}{180\div 36} \, radians = \frac{7\pi}{5} \, radians$$

Step 5: Write the final answer

So, the equivalent measure of $$252^{\circ}$$ in radians is $$\frac{7\pi}{5} \, radians$$.

Key Concepts

Angle Measurement

Before diving into converting degrees to radians, it's important to understand angle measurement itself. Angles measure the amount of turn between two lines that originate from the same point. There are different ways to express this measurement, with

• Degrees (°)
• Radians

being two of the most common.
The degree system, which is widely used in everyday scenarios, divides a full circle into 360 equal parts. Therefore, each degree is equivalent to \(rac{1}{360}\) of a full circle. This method is handy for smaller, more intuitive angle values, like those on a protractor.
Understanding angles in degrees allows you to appreciate how rotation works and is essential in fields such as navigation and astronomy.

Radian Conversion

Radians provide another way of measuring angles, primarily used in mathematics and engineering. This might seem complex at first, but it's a powerful way to express angles.
Instead of dividing a circle into 360 parts like degrees, radians use the circle's radius as a natural unit of measurement. In a full circle, there are \(2\pi\) radians. This is due to the circumference of a circle being \(2\pi r\).
When converting degrees to radians, you use the conversion factor:

• \(1^{\circ} \approx 0.01745\, radians\)
• Therefore, \(252^{\circ}\) is obtained as \(\frac{252\pi}{180}\, radians\)

Understanding this conversion helps when dealing with mathematical problems requiring precision.

Trigonometry

Trigonometry is a branch of mathematics that involves studying triangles and relationships between their angles and sides. In trigonometry, angles are often measured in radians to facilitate the calculations in functions like sine, cosine, and tangent.
One of the key principles of trigonometry is that a circle measures \(2\pi\) radians around, simplifying many formulas. For example, when working with circles or periodic functions, using \(\pi\) makes understanding and solving these problems more intuitive.
For trigonometric functions, using radians instead of degrees helps connect angles directly to calculations involving arcs and circles, improving the accuracy and simplicity of expressions.

Mathematical Conversion

Mathematical conversion, especially between different units of measurement, is a vital skill. It forms the basis for correctly interpreting and solving various mathematical and real-world problems.
When converting degrees to radians, you follow a few simple steps:

• Write the degree measurement and understand the radian factor.
• Multiply the degree measure by the factor, \(\frac{\pi}{180}\), to convert to radians.
• Keep your answer as a fraction with \(\pi\), unless otherwise required.
• Simplify the fraction if possible, ensuring the most economical representation.

This conversion allows more meaningful interaction with equations and situations relying on angular measurements, especially in advanced math and physics.