Problem 32 - Contemporary Precalculus

Question

Convert the given degree measure to radians. $$-105^{\circ}$$

Step-by-Step Solution

Answer

Question: Convert the given degree measure $$-105^{\circ}$$ to radians. Answer: $$-\frac{7\pi}{12}$$ radians

Step 1: Understand the conversion factor between degrees and radians

To convert a given degree measure to radians, we need to use the conversion factor: $$1^{\circ} = \frac{\pi}{180} \space radians$$.

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

Now, let's multiply the given degree measure, $$-105^{\circ}$$, by the conversion factor, to convert it to radians: $$-105^{\circ} \cdot \frac{\pi}{180} = -\frac{105\pi}{180}$$

Step 3: Simplify the fraction

The final step is to simplify the fraction: $$-\frac{105\pi}{180}$$ We can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 15: $$-\frac{105\pi ÷ 15}{180 ÷ 15} = -\frac{7\pi}{12}$$ The given degree measure, $$-105^{\circ}$$, expressed in radians is $$-\frac{7\pi}{12}$$ radians.

Key Concepts

Understanding Radians

Radians are a unit of angular measure used in mathematics and engineering. Unlike degrees, which divide a circle into 360 equal parts, radians use the radius of the circle as the basis for measurement. This means one radian is the angle formed when the arc length equals the radius length of the circle.

• A full circle in terms of radians is represented as \(2\pi\) radians, which is equivalent to 360 degrees.
• Therefore, half a circle, or 180 degrees, equals \(\pi\) radians.

Using radians can greatly simplify many mathematical equations, especially those involving periodic functions like sine and cosine. They also provide a direct link between the length of an arc and the radius of the circle.

Concept of Degrees

Degrees are a widely-used unit for measuring angles, where a circle is divided into 360 equal parts. Each part is a degree, making it a practical system for everyday use and in various fields like geometry and trigonometry.

• A right angle is 90 degrees, representing a quarter turn around a point.
• Commonly, degrees are represented with the symbol \(^{\circ}\).
• Degrees are most frequently used in geographic navigation and in construction.

Degrees provide an easy-to-understand framework for angle measurement, making them very accessible for beginners learning about angles and geometry.

Using Conversion Factors

A conversion factor is a numerical factor used to change measurements from one unit to another. When it comes to angles, converting between degrees and radians is a common task. The conversion factor between these units is derived from the relationship between the circumference of a circle and its diameter, embodied in the value of \(\pi\).

• To convert from degrees to radians: multiply the degree measure by \(\frac{\pi}{180}\).
• To convert from radians to degrees: multiply the radian measure by \(\frac{180}{\pi}\).

Let's look at an example: if you have \(-105^{\circ}\), you multiply by \(\frac{\pi}{180}\) to convert to radians. The calculation gives you \(-\frac{7\pi}{12}\), showing how essential this factor is for accurate conversions.