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Pre-Calculus $\longrightarrow$
Trigonometry $\longrightarrow$

Objectives
• Learn the definition of radians and understand why it is a measure without dimension
• Translate between common degree measure angles and their equivalent radian measure
• Revisit measuring angles in the coordinate plane but this time using radians
• Become an expert on measuring angles with radians instead of degrees - you'll need it for the future!
Lesson Description

For much of the ride ahead through Trigonometry (and very much so in Calculus), we aren't able to use degrees as a way to measure angles. This lesson introduces the very important alternative, dimension-less way to measure angles, measured in units we call radians.

Practice Problems

Practice problems and worksheet coming soon!

## Degrees Aren't Good Enough

Up to this point, we've used degrees to measure the size of angles. A degree is a somewhat arbitrary unit that was created long before modern mathematics, and as we know, it is defined to be $1/360$ of a circle. However, it fails us as an angle measurement system when we get into advanced trigonometry and early Calculus, because it is what we call a dimensioned unit of measure.What's so wrong with that? Let's look at another dimensioned measure that we're also familiar with - distance. When you measure the distance of a hallway, you might measure it in feet. However, if you want to measure how much area the wall of that hallway has, you multiply its length by its height. What unit of measurement do you get if you measured both in feet? Feet squared. When you multiply or divide dimensioned units, the units of your answer change too.This is the problem with degrees. If we solve for $x$ that is a variable angle measurement measured in degrees, how do you interpret the units of $x^2$? Squared degrees doesn't mean anything, unlike squared feet.

To address this problem, mathematicians came up with a dimension-free way to measure angles in the coordinate plane. Instead of using a fixed, pre-defined unit, angles are measured as a fraction. That way, any unit of measure cancels with itself, and as a unit-less number, if you take the angle measurement and square it, square root it, or do anything else to it, it remains a unit-less number.The number is based on central angles and their corresponding sector arc length. The angle measurement is the number of radius lengths that the sector arc length takes up. Let's define and understand this properly.
Radian Angle MeasurementAn angle's measurement in radian measure is equal to the arc length of the central angle's circle sector, measured by number of circle radius lengths. Like degrees, this measurement is universal and does not depend on the size of the circle you use for the sector, because any two sectors with the same angle have the same arc length measurement when measured in number of radius arc lengths If the circle enlarges, so does the radius length.However, as it turns out, an angle measured with an integer number of radians will never be a "nice" angle. In fact, you'll never work with angles like $2$ or $3$ radians throughout your trig studies (except for rare questions that are usually trick questions).Instead, the common angles you'll work with will be multiples of $\pi$ and fractions of $\pi$. The underlying reason is that the number of radius lengths that wrap around a whole circle is not an integer - it is $\pi$ times $2$.Another way to say that is, there are $\pi$ radius lengths in the arc length of a semi-circle: Just because we need to start using radians doesn't mean the usual angle measures will stop being popular - like $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, and $90^{\circ}$. Instead, we simply have to get used to their "radian names."A moment ago we commented that there are $\pi$ radius lengths around the arc length of a semi-circle. This means that the central angle of a semi-circle is $\pi$ radians. However, we also know that a semi-circle angle, when measured in degrees, is $180^{\circ}$. We conclude that $180^{\circ}$ is equivalent to $\pi$ radians.
Converting Between Degrees and RadiansIn order to convert an angle measured in degrees to radians, multiply the measurement by$$\frac{\pi}{180^{\circ}}$$In order to convert an angle measured in radians to degrees, multiply the measurement by$$\frac{180^{\circ}}{\pi}$$
I Used To Know That! There are $360^{\circ}$ in the central angle that spans exactly once around a circle.
You Should Know In Chemistry we're taught how to use dimensional analysis to convert quantities from one set of unit measurement to another, such as liters of gas to number of molecules. While we don't need to revisit that stuff explicitly, there is a huge benefit to remembering that concept here - it helps you keep track of which number should be on top when converting between degrees and radians or vice versa.Converting degrees to radians uses $\pi/180^{\circ}$ as the multiplier because the units cancel. E.g.$$45 \; \cancel{\mathrm{deg}} \cdot \frac{\pi \; \mathrm{rad}}{180 \; \cancel{\mathrm{deg}}} = \frac{\pi}{4} \; \mathrm{rad}$$Converting radians to degrees uses the reciprocal multiplier to cancel the appropriate units.

Example 1Convert $36^{\circ}$ to radians.$\blacktriangleright$ Let's multiply by the appropriate multiplier, and then simplify.$$36^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{\pi}{5}$$

Example 2Convert $5\pi / 12$ radians to degrees.$\blacktriangleright$ In this case, multiply by $180/ \pi$:$$\frac{5\pi}{12} \cdot \frac{180^{\circ}}{\pi} = \frac{900 \pi}{12 \pi}^{\;\circ}$$$$=75^{\circ}$$

In the last lesson » we learned how to plot standard position angles in the plane. While we went over how this works using degrees, many teachers want you to be able to handle all standard position related tasks using either degrees or radians.Common AnglesThe angles that we get by dividing a circle into integer pieces, such as $90^{\circ}$, $60^{\circ}$, $45^{\circ}$, etc. are angles we will work with often. While you can always use the conversion fractions to go back and forth between these degree-measured angles and their radian form, it will be a huge boon to know how these angles and their multiples automatically.The first part of the pattern is that these angles look like fairly simple fractions of $\pi$ when represented in radian measure. This happens because there are $2\pi$ radians in a full circle, so, for example, dividing a circle into $8$ equal pieces where each piece has a $45^{\circ}$ angle is an angle that is equivalent to dividing $2\pi$ into $8$ pieces:$$\frac{2\pi}{8} = \frac{\pi}{4}$$These are the acute common angles you essentially will effortlessly memorize due to the frequency with which you will see them:\begin{align} 30^{\circ} & \longrightarrow & \frac{\pi}{6} \\ \\ 45^{\circ} & \longrightarrow & \frac{\pi}{4} \\ \\ 60^{\circ} & \longrightarrow & \frac{\pi}{3} \\ \\ 90^{\circ} & \longrightarrow & \frac{\pi}{2} \end{align}Finally, know that multiples of these angles in Quadrants $2$, $3$, and $4$ have the same denominators.For example, if $\pi / 6$ is $30^{\circ}$, then $5\pi / 6$ is $5 \times 30^{\circ}$ or $150^{\circ}$.
Pro Tip When you need to convert larger angles to radians, you can use multiples of the acute angles to make it easier. For example, $330^{\circ}$ is $11 \times 30^{\circ}$ so $330^{\circ}$ is $11\pi / 6$ in radian measure.
Co-Terminal AnglesFinding co-terminal angles when working in radians is conceptually the same - you add or subtract a full circle. Where degree based calculations use $360^{\circ}$ for this task, radian based calculations will use $2\pi$.

Example 3Find a positive and negative co-terminal angle to $7\pi / 6$.$\blacktriangleright$ We don't need to convert this angle to degrees or even know where it lands in the plane to complete this task. Let's just add and subtract $2\pi$ to get each answer.Positive co-terminal:$$\frac{7\pi}{6} + 2\pi$$$$= \frac{7\pi}{6} + \frac{12\pi}{6}$$$$= \frac{19\pi}{6}$$Negative co-terminal:$$\frac{7\pi}{6} - 2\pi$$$$= \frac{7\pi}{6} - \frac{12\pi}{6}$$$$= -\frac{5\pi}{6}$$As you can see, the key is to turn $2\pi$ into whatever denominator you are working with.Note that we only needed to add and subtract $2\pi$ once for each answer because the angle was a "typical" positive angle that is not more than one rotation. Otherwise we would need to add or subtract $2\pi$ multiple times.

Negative AnglesNothing new here - just be careful counting backwards (that is, clockwise) in radians because it can be disorienting when you're first learning it.

As you'll soon see, it is important that you can quickly identify which quadrant a given radian measured angle is in. While you could convert it to degrees which is more familiar, it is advantageous to be able to quickly identify quadrants without that unnecessary step.The trick is to use fractions of $\pi$ as a guide. Since you know that $180^{\circ}$ is equal to $\pi$, take any fraction of $\pi$ angle and figure out whether it is larger than or smaller than $\pi$, and then go from there. For larger angles it is sometimes helpful to compare to $2\pi$.

Example 4Determine what quadrant the following four angles are in.a) $7\pi/4$b) $2\pi/3$c) $6\pi/5$d) $13\pi/100$$\blacktriangleright Use \pi = 180^{\circ}, \pi/2 = 90^{\circ}, and 2\pi = 360^{\circ} as guides when considering fractions of \pi.a) 7\pi/4 is just \pi/4 shy of a full circle of 2\pi (in this case, think 8 \pi / 4 for 2 \pi), so we know it's in quadrant 4.b) 2/3 is bigger than 1/2 but smaller than 1, so 2\pi/3 must be in the second quadrant since \pi / 2 is 90^{\circ} and \pi is 180^{\circ}.c) 6/5 is a mere 1/6 more than 1, so 6\pi/5 is only slightly larger than \pi and must be in the third quadrant.d) 13\pi/100 is between 0 and \pi/2, and thus it lies in quadrant 1. ## Mr. Math Makes It Mean Because we work with common angle measures 99\% of the time, teachers like to throw odd questions on quizzes that involve converting uncommon angles.Random DegreesInstead of converting multiples of 15^{\circ}, you might be asked to convert something like 7^{\circ} to radians. The process isn't any different but the answers will look strange compared to what you are used to seeing because the fractions will have large denominators. Example 5Convert 7^{\circ} to radians.\blacktriangleright As always, accomplish this by multiplying by \pi / 180:$$7^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{7\pi}{180^{\circ}}$$This answer cannot be further simplified. The prior example wasn't overly difficult to execute, it's just that you'll get used to questions that give familiar answers and you may feel like something is wrong when asked an unusual question.That is probably even more true of unusual conversions that go the other way (from radians to degrees), as we'll see in the next example. Example 6Convert 4 radians to degrees.\blacktriangleright Again, mechanically there is no change to our approach. We should multiply by 180/ \pi.$$4 \cdot \frac{180^{\circ}}{\pi} = \frac{720}{\pi}^{\;\circ}$$The proper, exact answer is 720 / \pi^{\,\circ} and cannot be simplified or changed.We could use a calculator to approximate this as 229.18^{\circ}. We should not expect an integer result since 1 radian is not an integer number of degrees. ## Put It To The Test Example 7Approximate the number of degrees in 1 radian. Show solution \blacktriangleright To accomplish this task, let's start by converting 1 radian to degrees with the conversion process:$$1 \cdot \frac{180^{\circ}}{\pi} \approx 57^{\circ}$$One radian is 180 / \pi which a calculator approximates as 57.2958, or 57 rounded to the closest degree. Example 8Convert 135^{\circ} to radians. Show solution \blacktriangleright$$$135^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{3 \pi}{4}$$Example 9Convert 75^{\circ} to radians. Show solution \blacktriangleright$$$75^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{5 \pi}{12}$$Example 10Convert 2\pi / 3 to degrees. Show solution \blacktriangleright$$$\frac{2 \pi}{3} \cdot \frac{180^{\circ}}{\pi} = 120^{\circ}$$Example 11Convert 11\pi / 18 to degrees. Show solution \blacktriangleright$$$\frac{11 \pi}{18} \cdot \frac{180^{\circ}}{\pi} = 110^{\circ}$$Example 12Find a positive and a negative co-terminal angle to 8\pi / 3. Show solution \blacktriangleright First let's subtract 2 \pi:$$\frac{8 \pi}{3} - 2\pi \longrightarrow \frac{8 \pi}{3} - \frac{6 \pi}{3} = \frac{2 \pi}{3}$$Here, we have found a co-terminal positive angle by subtracting because our original angle 8 \pi / 3 was already larger than one full circle rotation. We could have added 2 \pi and got a positive co-terminal angle, but this way we're closer to the other half of the answer we need. Let's subtract 2 \pi once more to obtain the negative co-terminal angle.$$\frac{2 \pi}{3} - \frac{6 \pi}{3} = - \, \frac{4\pi}{3}$$Therefore, an angle that is positive and co-terminal with 8 \pi / 3 is 2 \pi / 3, and an angle that is negative and co-terminal with 8 \pi / 3 is -4 \pi / 3. There are many other correct ways to answer the question as well. Example 13Convert 28^{\circ} to radians. Show solution \blacktriangleright$$$28^{\circ} \cdot \frac{\pi}{180^{\circ}} = \frac{7 \pi}{45}$$Example 14Convert 3.6 radians into degrees. Show solution \blacktriangleright$$$3.6 \cdot \frac{180^{\circ}}{\pi} = \frac{648}{\pi}^{\circ}$$Remember that when you work with a number of radians that is not a fraction of$\pi$, you will get$\pi\$ in the denominator of your answer.

Lesson Takeaways
• Understand what radians are and how we use them
• Be able to translate back and forth between degrees and radians
• Know how to quickly identify the quadrant of any radian measured angle
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Pro-Tip: Knowing these will make your life easier.

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