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## Trigonometry

### Course: Trigonometry>Unit 2

Sal discusses the general approach to converting between radians and degrees and vice versa. Created by Sal Khan.

## Want to join the conversation?

• Can you possibly have negative angles?
• Negative angles are clockwise angles. (Counterclockwise is positive)
• If pi continues forever, how can we use it to define answers? That would mean every answer we get would continue on forever, but we shorten pi and thus makes none of the math we do with pi actually 100% true but rather an estimated amount. I don't even understand the concept of pi honestly. Can someone explain to me?
• Take a measurement of a length of anything, we won't get an exact whole number. A pencil said to be 8 cm long may be 8.000034 cm for example. We are always estimating because the exact amount is almost never needed, and we take as accurate a measurement as required. So, every answer may continue on forever, but what we estimate, is what we need practically.

Theoretically in math, since we always use rational numbers most of the time, an irrational number like pi is often confusing as it does not provide a definite rational answer. Instead we have to estimate to the accuracy required for the situation. If you want to find the circumference of a random cart wheel, you dont need accuracy. When you find the circumference of a rocket, you may need more accuracy.

But of course, theoretically we can still get a definite answer if we just dont expand π and leave it as π. Circumference of a circle of diameter 3 is 3π. This gives you a perfect theortical answer. Otherwise it would be 3*3.14..... and as you said, it is not a perfect defined answer and is not theoretically accurate.

Hope this helps!
- Super7SonicX
• Is there any kind of notation for radians?
• Yes, there is, though it is rarely used.
You write degrees with a little circle at the top 1.2°
Same way, an angle of 1.2 radians would be written either as "1.2 rad" or "1.2 with a "c" at the top.(I can't seem to get the 'c' using formatting here.)

See-
• Are negative degrees actual things, or are they hypothetical like negative numbers?
• they are actual things. For example, if you rotate an object 90 degrees clockwise, it would be -90 degrees. Like the number line, negative and positive only show direction
• Is there any other way to measure the angle just like degrees, radians....?
• There was an attempt at a metric measure of angle where the right angle was divided into 100 parts (as opposed to the usual 90 degrees). The measure was called the gradian. There were 400 gradians in a complete revolution, and 1 gradian = 0.9 degrees.

It hasn't really caught on, and the only place I've seen it is on calculators.

Is that what you had in mind?
• Is 1.5 pi the same as 270?
• Yep, 1.5π radians is exactly 270°.

We usually use fractions for radians, so that would be 3π/2. What you said is completely correct, though!
• Why did humans invent radians and degrees? Isn't one enough?
• Radians make calculation easier in dealimg with derivatives.

For example, if you take the derivative of sin x that will be
pi/180 cos x using degrees however by defining pi=180 the derivative will just be cos x which is simpler. You can get the result from the proof theorem on the derivative of sin x being cos x except instead of using radians as Sal does in his calculations use degrees. You will also notice that:

lim_x->0 sin x/x doesn`t equal 1 but pi/180 from using degrees by following the steps he carries out.

To understand the proof you should, however, have an understanding of limits/differentiation and circular geometry has in finding arc length and the area of a sector which you can learn about in some of Sal videos.
• When I have 5pi/9 and I am converting it to degrees do I set them up in the proportion just as in the video? And just like the example with -pi/2 do I multiply the 5 by 180? My answer was 100 degrees. Would like to know if I did it right. thx!!
• You are correct, 5π/9 * 180/π = 100. Notice you could multiply 5 * 180 and divide by 9 or you could divide 180/9 to get 20 and then multiply by 5 to get 100.
• Why is the coefficient for radians usually irrational but coefficients for degrees are rational?
• This could be because we are using pi as part of our way to convert between the 2. Also, if you watch "Intro to Radians", you'll find that the original degree system was based off of whole numbers(could've played a role as well).
• This is really ... Extremely confusing,What are radians?
• Radians got me for years too, but once you understand them it actually makes more sense than the degrees system, specifically when talking about circles. The degrees system is based on some Babylonian dude thousands of years ago deciding that an angle of one degree in the center of the circle brings you exactly 1/360th around the edge of the circle, hence a circle having 360 degrees. It just kind of stuck. Radians are simply how many radiuses (yeah, I know it's radii) of the circle it takes to get all the way around, which is 2pi or (2xpi=6.28), so it takes 6.28 radians to get all the way around the circle and back to where you started. One radian is about 57 degrees. Degrees feel comfortable right now because we grew up using them and they're useful for some calculations but the more math you do, the more you will prefer radians. Hope this helped.