Problem 13 - Precalculus Mathematics for Calculus

Question

Find the radian measure of the angle with the given degree measure. $$7.5^{\circ}$$

Step-by-Step Solution

Answer

The radian measure is \( \frac{\pi}{24} \).

Step 1: Understand Radians and Degrees

The radian is a unit of angular measure used in mathematics. To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This represents the ratio of radians to degrees since \( 180^{\circ} = \pi\) radians.

Step 2: Set Up the Conversion Formula

To convert \( 7.5^{\circ} \) to radians, use the formula:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]So, substitute \( 7.5 \) for Degrees:

Step 3: Calculate the Radian Measure

Substitute the degree value into the formula:\[7.5 \times \frac{\pi}{180} = \frac{7.5\pi}{180}\]Now simplify the fraction by dividing both the numerator and the denominator by 7.5:

Step 4: Simplify the Fraction

Divide both 7.5 and 180 by 7.5:\[\frac{7.5\pi}{180} = \frac{\pi}{24}\]Thus, the simplest form of the radian measure for \( 7.5^{\circ} \) is \( \frac{\pi}{24} \).

Key Concepts

Radian Measure

When we talk about radian measure, we're discussing an alternative way to describe angles, different from degrees. Imagine a circle: a radian is the angle created when the arc length is equal to the radius of the circle. This forms a natural unit of angle measure and ties directly to the properties of the circle. One full circle is equivalent to an angle of \(2\pi\) radians. This is because the circumference of a circle is \(2\pi\) times its radius. So, for the angle you have, when we convert from degrees, we're simply translating into this circle-based measurement.

Angular Measure

Angles can be measured in two main units: degrees and radians. Degrees are perhaps what most people are familiar with—a full circle is \(360^{\circ}\). Radians, on the other hand, are used frequently in mathematics due to their relationship with the circle's radius. The choice between using degrees or radians depends on the context. If you're dealing with geometric problems on paper, degrees might be more intuitive. However, in calculus and higher mathematics, radians offer a more natural fit because they simplify many formulas and calculations.
Understanding the relationship between these two units is crucial, as it helps switch between them seamlessly.

Conversion Factor

Converting between degrees and radians requires a specific conversion factor, essentially acting like a translator between these units. The conversion factor is \(\frac{\pi}{180}\), because \(180^{\circ}\) equals \(\pi\) radians. This factor allows you to convert any angle from degrees to radians by multiplying it by \(\frac{\pi}{180}\).

• To convert from degrees to radians: Multiply by \(\frac{\pi}{180}\).
• To convert from radians to degrees: Multiply by \(\frac{180}{\pi}\).

This factor is simple but incredibly powerful, giving you a straightforward way to switch units whenever necessary.

Simplifying Fractions

Once you apply the conversion factor, you often get a fraction that can be simplified. In the given exercise, after multiplying \(7.5\) by \(\frac{\pi}{180}\), you end up with the fraction \(\frac{7.5\pi}{180}\). Simplifying fractions involves dividing both the numerator and the denominator by their greatest common divisor.

• Here, you divide both terms by \(7.5\), simplifying \(\frac{7.5\pi}{180}\) to \(\frac{\pi}{24}\).
• This step ensures the radian measure is in its simplest form, making it much easier to work with in further calculations.

Knowing how to simplify fractions is a key math skill. It allows answers to be more concise and often reveals more about the relationships between numbers.