Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Leave answers as rational multiples of \(\pi .\) $$-210^{\circ}$$

Step-by-Step Solution

Answer

The degree measure \(-210^{\circ}\) converts to \(-\frac{7\pi}{6}\) radians.

Step 1: Understanding the Conversion Factor

To convert degrees to radians, use the conversion factor: \[1^{\circ} = \frac{\pi}{180^{\circ}} \] This formula tells us how many radians correspond to each degree.

Step 2: Multiply the Degree Measure by the Conversion Factor

Start with the degree measure \(-210^{\circ}\) and multiply it by the conversion factor for radians: \[-210^{\circ} \times \frac{\pi}{180^{\circ}} = -\frac{210\pi}{180}\] This sets the stage for simplifying the fraction.

Step 3: Simplify the Fraction

The fraction \(-\frac{210\pi}{180}\) can be simplified by finding the greatest common divisor of 210 and 180. Both numbers are divisible by 30:\[-\frac{210 \pi}{180} = -\frac{210 \div 30 \pi}{180 \div 30} = -\frac{7\pi}{6}\] As a result, the simplified form of the radian measure is \(-\frac{7\pi}{6}\).

Key Concepts

Radians

Radians are a way to measure angles, just like degrees, but they are based on the properties of circles. While degrees divide a circle into 360 equal parts, radians offer a more natural fit in mathematical calculations. Think of radians as a ratio that involves the arc length of a circle and its radius. This connection makes radians particularly useful in trigonometry and calculus.
If you think of the circumference of a circle, it is expressed in terms of \(2\pi r\) where \(r\) is the radius. Since the circle is one complete revolution, this gives us the fundamental reason \(2\pi\) radians equals \(360^{\circ}\). Hence, \(\pi\) radians represents \(180^{\circ}\), creating that critical conversion link between the two units.

Conversion Factor

The conversion factor is a pivotal tool to shift seamlessly between degrees and radians. The conversion factor linking degrees to radians is \(1^{\circ} = \frac{\pi}{180}\). This indicates that for every degree, you have \(\frac{\pi}{180}\) radians. It's this relationship that allows us to exchange degree measures for radian form.
Using the conversion factor is pretty straightforward. Here's a simple guide:

• Start with your angle in degrees.
• Multiply the degree by the conversion factor \(\frac{\pi}{180}\).
• The result will be in radians.

This conversion is essential especially when dealing with trigonometric functions and calculus concepts, where radians are often preferred.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This is essential to make expressions cleaner and easier to interpret, especially in mathematics. When converting degrees to radians, you'll often end up with a fraction containing \pi\. Simplifying involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, take the conversion of \(-210^{\circ}\) to radians, which gives \(-\frac{210\pi}{180}\). The GCD of 210 and 180 is 30, so divide both by 30:

• \(210 \div 30 = 7 \)
• \(180 \div 30 = 6 \)

This yields the simplified fraction \(-\frac{7\pi}{6}\), making it easier to use in further calculations.

Trigonometry

Trigonometry is a branch of mathematics focused on the study of triangles, particularly right triangles, and the relationships between their angles and sides. This field extensively uses radians because they simplify many mathematical concepts and calculations.
In trigonometry, radians allow for more seamless integration with calculus, facilitating the use of derivative functions and integrals. Functions like sine, cosine, and tangent often operate more naturally in radian measure due to their cyclic properties.

• Using radians can make understanding the periodic nature of trigonometric functions more intuitive.
• Angles expressed in radians ensure that formulas result in neat, exact values.

Thus, transitioning to radians when dealing with trigonometric calculations is a common and effective practice.