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Chapter 5: Problem 4

Convert each angle to radians. \(-60^{\circ}\)

Short Answer

Expert verified
The angle \(-60^{\circ}\) in radians is \(-\frac{\pi}{3}\).

Step by step solution

01

Recall the conversion factor between degrees and radians

Recall that 1 radian = \(180^{\circ}/\pi\). This conversion factor allows us to convert an angle given in degrees to radians by multiplying it with the inverse of the conversion factor, i.e., \(\pi / 180^{\circ}\).
02

Use the conversion factor to convert the given angle to radians

Now, to convert the given angle "\(-60^{\circ}\)" to radians, multiply it by the inverse of the conversion factor, that is: \(-60^{\circ} \times \frac{\pi}{180^{\circ}}\)
03

Simplify the expression to get the angle in radians

Simplify the expression: \(-60^{\circ} \times \frac{\pi}{180^{\circ}} = -\frac{60\pi}{180}\) Now, we can simplify further by dividing the numerator and denominator by their greatest common divisor (GCD), which is 60: \(-\frac{60\pi}{180} = -\frac{1\pi}{3}\) So, the given angle "\(-60^{\circ}\)" in radians is \(-\frac{\pi}{3}\).

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Key Concepts

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

Radians
Radians are a way to measure angles, just like degrees. However, radians are based on the radius of a circle. Imagine wrapping the radius of a circle along its circumference. The angle subtended by this arc length is 1 radian.
In simpler terms, a radian measures how much of the circle's radius fits into the circle's circumference. Since the entire circumference of a circle is \(2\pi\) times the radius, there are a total of \(2\pi\) radians in a full circle.
This is why you often see \(\pi\) in radian values: it represents half a circle, or 180 degrees.
Degrees
Degrees are another way to measure angles and are probably more familiar to most people. A full circle is divided into 360 degrees, which makes it easy to understand and visualize simpler angles.
One advantage of using degrees is that they are divided into smaller units like minutes and seconds, which can be helpful for precise calculations.
When working on problems involving circles or circular motion, degrees often seem intuitive. But radians provide a more natural measurement when dealing with mathematics and physics.
Conversion Factor
The conversion factor is a handy tool for switching between radians and degrees. Knowing the relationship between these two measurements is essential.
To convert from degrees to radians, multiply the angle in degrees by \(\frac{\pi}{180}\)\. This factor comes directly from the relationship where \(180^{\circ} = \pi\) radians.
Conversely, to convert from radians to degrees, you would multiply by the reciprocal conversion factor, \(\frac{180}{\pi}\). With these conversion factors in hand, you can easily switch back and forth between radians and degrees in any problem.

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Most popular questions from this chapter

Evaluate \(\sin \left(-\sin ^{-1} \frac{3}{13}\right)\).

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with \(\tan u=-2\) and \(\tan v=-3\) Find exact expressions for the indicated quantities. $$ \sin (-u) $$

Without using a calculator, sketch the unit circle and the radius that makes an angle of \(\cos ^{-1} 0.1\) with the positive horizontal axis.

Is arctangent an even function, an odd function, or neither?

Explain why $$ \cos ^{-1} \frac{5}{13}=\sin ^{-1} \frac{12}{13}=\tan ^{-1} \frac{12}{5} $$.
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