Problem 11 - Precalculus Mathematics for Calculus

Question

Find the radian measure of the angle with the given degree measure. $$96^{\circ}$$

Step-by-Step Solution

Answer

The angle is \( \frac{8}{15} \pi \) radians.

Step 1: Understanding the Conversion Formula

To convert from degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula is derived from the fact that \( 180^{\circ} \) is equivalent to \( \pi \) radians.

Step 2: Applying the Conversion Formula

Plug in the given degree measure into the conversion formula. Here, we substitute \( 96^{\circ} \) for degrees in the formula: \( 96 \times \frac{\pi}{180} \).

Step 3: Simplifying the Expression

Now, we simplify the fraction. First, divide 96 by 180: \( \frac{96}{180} \). Simplify this fraction: both 96 and 180 are divisible by 12, so \( \frac{96}{180} = \frac{8}{15} \).

Step 4: Expressing in Radians

Multiply the simplified fraction by \( \pi \): \( \frac{8}{15} \pi \). Therefore, \( 96^{\circ} \) is \( \frac{8}{15} \pi \) radians.

Key Concepts

Angle Measurement

In mathematics, measuring angles is a crucial skill. Angles can be measured in different units, which are used in various applications such as geometry, trigonometry, and even physics.

When we talk about angle measurement, the most commonly used units are degrees and radians. Degrees are familiar in everyday contexts, like telling time or navigating directions. A full circle is measured as 360 degrees.

When we divide it into four quadrants, each section consists of 90 degrees, such as with a right angle. This unit is quite intuitive, making it easy to visualize. However, degrees are not very practical in advanced mathematics because calculations involving degrees can be cumbersome. Radians, on the other hand, provide a more natural and efficient way to measure angles relative to the circle’s radius, especially when dealing with trigonometric functions.

Radian Measure

The radian is a unit of angular measure that is highly valuable in advanced math and sciences.

A key aspect to understand is that the radian measures the angle produced when the length of the arc it subtends on a circle is exactly equal to the circle's radius.Imagine a circle with a radius of 1 unit. If you draw an arc of 1 unit along the circumference of the circle, the angle that arc creates at the circle's center is 1 radian. Because of this relationship to the radius, radians connect nicely with many mathematical formulas, including those that involve trigonometric functions.

Another point to note is the connection of radians to the ratio of the circle's circumference. As the circumference of a circle is 2π times its radius, a full circle has 2π radians, making a half circle π radians. This relationship makes radians ideal for analytical computations, where pi (\(π\)) frequently appears.

Conversion Formula for Angles

Converting between degrees and radians is a common task in mathematics, and it uses a simple formula. The conversion formula stems from the equivalence that 180 degrees is equal to π radians.

This is essential because it forms the basis for converting any angle from degrees to radians. To convert an angle from degrees (\(^{ ext{o}}\)) to radians, you multiply the degree measure by \(\frac{\pi}{180}\).

Here's the formula for easy reference:

• Radians = Degrees × \(\frac{\pi}{180}\).

For example, to convert 96 degrees to radians, you substitute 96 into the formula:

• \(96 imes \frac{\pi}{180} = \frac{8}{15} \pi\)

This method is not only efficient but also ensures accuracy, providing a straightforward way to switch between these two important units of measure.