Problem 26 - Precalculus Mathematics for Calculus

Question

Find the degree measure of the angle with the given radian measure. $$-\frac{13 \pi}{12}$$

Step-by-Step Solution

Answer

The degree measure of the angle is \(-195\) degrees.

Step 1: Understand the Conversion

One full circle (360 degrees) is equivalent to \(2\pi\) radians. Therefore, to convert from radians to degrees, we can use the conversion factor \( \frac{360}{2\pi} = \frac{180}{\pi} \).

Step 2: Apply the Conversion Factor

Multiply the given radian measure \(-\frac{13\pi}{12}\) by \( \frac{180}{\pi} \) to convert it into degrees.\[-\frac{13\pi}{12} \times \frac{180}{\pi}\]

Step 3: Simplify the Expression

Simplify the expression by cancelling \(\pi\) and performing the multiplication:\[= -\frac{13 \times 180}{12}\]Calculate the multiplication and the division.

Step 4: Perform the Arithmetic

Carry out the arithmetic:\[= -\frac{2340}{12} = -195\] Thus, the angle in degrees is \(-195\) degrees.

Key Concepts

Radian to Degree Conversion

Did you know that angles can be measured in both radians and degrees? Converting between these two units is a core skill in trigonometry. To understand this, imagine a circle. A full circle has 360 degrees, but in radians, it is equal to \(2\pi\). This means we have a handy conversion factor: \(\frac{180}{\pi}\). To convert a radian measure to degrees, you multiply the radian by this conversion factor.

• For example, if you have \(-\frac{13\pi}{12}\) radians, multiply by \(\frac{180}{\pi}\).
• Keep in mind that \(\pi\) will cancel out in the process.

Converting between these two units is essential since some problems are easier to solve in degrees while others in radians. Try to practice it often, as it will always come in handy!

Arithmetic Simplification

Arithmetic simplification is all about making complex expressions simpler or easier to understand. Once you've set up the equation for converting radians to degrees, it’s time to simplify it.

• First, cancel out identical terms to reduce the complexity; here,\(\pi\) cancels out.
• Next, perform the multiplication: multiply\(-13\) by \(180\) to get \(-2340\).
• Finally, carry out the division. Divide \(-2340\) by \(12\).

The result of this process is often a straightforward number, which is much easier to interpret. Simplification not only saves you time but also minimizes errors in solving more complex problems.

Trigonometry Basics

Trigonometry is a branch of mathematics that deals with angles, triangles, and their relationships. Understanding the basics of trigonometry is vital if you're working with angles, whether in radians or degrees.

• Angles are a fundamental unit in trigonometry. They can be measured in two main units: degrees and radians.
• Radians are particularly useful in calculus and more advanced mathematical studies.
• The trigonometric functions—sine, cosine, and tangent—are based on these angles and are used to solve various problems related to triangles and circles.

By mastering trigonometry basics, you’ll have a great foundation for further exploration into more complicated mathematical concepts. Whether converting angles, simplifying expressions, or solving trigonometric equations, understanding these concepts makes mathematics much more approachable and fun!