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Chapter 6: Problem 26

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

Short Answer

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

Step by step solution

01

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 \( \frac{360}{2\pi} = \frac{180}{\pi} \).
02

Apply the Conversion Factor

Multiply the given radian measure \(-\frac{13\pi}{12}\) by \( \frac{180}{\pi} \) to convert it into degrees.\[-\frac{13\pi}{12} \times \frac{180}{\pi}\]
03

Simplify the Expression

Simplify the expression by cancelling \(\pi\) and performing the multiplication:\[= -\frac{13 \times 180}{12}\]Calculate the multiplication and the division.
04

Perform the Arithmetic

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥攕ine, cosine, and tangent鈥攁re based on these angles and are used to solve various problems related to triangles and circles.
By mastering trigonometry basics, you鈥檒l 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!

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Most popular questions from this chapter

A parallelogram has sides of lengths 3 and 5 and one angle is \(50^{\circ} .\) Find the lengths of the diagonals.

Find the exact value of the trigonometric function. $$\tan \frac{5 \pi}{2}$$

Find the exact value of the expression. $$\tan \left(\sin ^{-1} \frac{12}{13}\right)$$

The custom of measuring angles using degrees, with \(360^{\circ}\) in a circle, dates back to the ancient Babylonians, who used a number system based on groups of \(60 .\) Another system of measuring angles divides the circle into 400 units, called grads. In this system a right angle is 100 grad, so this fits in with our base 10 number system. Write a short essay comparing the advantages and disadvantages of these two systems and the radian system of measuring angles. Which system do you prefer? Why?

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation \(\theta\) of a rainbow is always the same. It can be shown that \(\theta=4 \beta-2 \alpha,\) where $$\sin \alpha=k \sin \beta$$ and \(\alpha=59.4^{\circ}\) and \(k=1.33\) is the index of refraction of water. Use the given information to find the angle of elevation \(\theta\) of a rainbow. (For a mathematical explanation of rainbows see Calculus Early Transcendentals, 7th Edition, by James Stewart, page 282 ).
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