So we're going to look at radian measure. Now, some 2,000 years ago, degree measure—as in 60 degrees or 90 degrees or 30 degrees—was arbitrarily chosen as a unit of measure. The problem with this is because it was arbitrarily chosen, it doesn't really get along well with any of our other units of measure, whether that's a measure of time, as in minutes, seconds, hours, or distance, as in feet or inches or meters or kilometers or miles. It doesn't match. It doesn't work.
If you look at the equation right here in the middle of this page, f(x) = eXsin(3x), imagine if we wanted to evaluate this where X equals 30 degrees. Well, X in this kind of a function is intended to be a unit of time or a unit of distance. And so 30 degrees is not gonna make sense in this particular function. Yes, we could put it into sine and get a reasonable answer out, but then we would have to raise E to the 39th power and get some huge gigantic number that's just gonna blow up the function and not make any sense at all.
So we cannot really use degrees when we're working with functions. So we had to find some other kind of a measure that would work and would make sense and still measure an angle. And so what we came up with was the radian measure, and it's a unitless measure, so it can work with any other measure.
And if we look at this diagram right here, we can see what a radian really is. And in this case, this circle should be considered the unit circle, where this is positive one on the Y access, negative one on the X axis, negative one on the Y axis. So we have a radius of one all the way around. And essentially, what a radian is, is the portion of the unit circle that is cut off by an angle that intersects the unit circle with a radius of one. This radius can be anywhere around this unit circle. So we could have the radius here or here or here and each of those would be a different radian measure.
But essentially, the radian is the amount of the circle cut off, so that's what we're really getting at. And because the radian is more comparable to a distance 'cause it is part of the distance around a circle, it does correspond better to the other, more typical units that, again, we've talked about in terms of functions, a.k.a. units of time or units of distance. So here is the unit circle with all this information filled into it. And what you've got is you can see all the most typical radian measures coming around the circle beginning with a radian of zero, then pi over 6, then pi over 4, pi over 3, and so on. And over here we have 3pi over 4, 5pi over 6, and then we get to pi. So halfway around this unit circle we get pi. And then we keep going, and now we're bigger than pi. 4pi over 3, et cetera, and then we're way bigger than pi and we end up at 11pi over 6. And then we finish the circle.
Well, after 11pi over 6, we get 12pi over 6. And 12pi over 6 is equivalent to 2pi, so the entire circle then— the entire unit circle has a circumference of 2pi, and each radian is some portion of the circumference of the entire circle. You can also see here the equivalent degree measures. So 45 degrees is equal to pi over 4. Two hundred and forty degrees is equal to 4pi over 3; 270 degrees is 3pi over 2; and so on.
You can also see in these points—and these points represent the values you would get for cosine and sine. Remember from our previous learning that cosine is associated with the X coordinate and sine is associated with the Y coordinate. So if I ask for the sine of pi over 3, I would want to get the square root of 3 over 2. If I ask for the cosine of pi over 3, that would be the X, or one-half. If I wanted the sine of 4pi over 3, that would be negative square root 3 over 2 because that's the X coordinate. If I wanted the cosine of 4pi over 3, I would want to negative one-half because that is the X coordinate of this point. And particularly, if I know these three, 30—these three: pi over 6, pi over 4, and pi over 3 and their associated points—then I can find the measure of any of their multiples.
So suppose that I wanted to find the cosine of 315 degrees. Well, 315 degrees is a multiple of 45. And 315 degrees is over here in the fourth quadrant. And in the fourth quadrant, sine is negative. So we know that the sine of pi over 4 or 45 degrees is square root of 2 over 2, but in the fourth quadrant sine is negative, so therefore, the sine of 315 degrees has got to be negative square root of 2 over 2. And if you actually look at that, lo and behold, the Y coordinate of the point associated with 315 degrees is negative square root of 2 over 2.
If I wanted to find the cosine of 5pi over 6, while the cosine of pi over 6 is square root of 3 over 2, 5pi over 6 has got to be in the second quadrant 'cause it's not bigger than pi yet. And in the second quadrant, cosine values are negative. So if the cosine of pi over 6 is square root of 3 over 2, then the cosine of 5pi over 6, which is in the second quadrant, has got to be negative square root of 3 over 2, and we, in fact, see that in the point over here.
So the point is, is that if we know all of the first quadrant radian measures and their associated points, we can find then the values of all the radian measures in all the other quadrants, which is very nice.
So let's talk a little bit now about how we convert from radian measure to degree measure and vice-versa. So let's begin by talking about how we convert from degree measure to radian measure, and to do that, we simply multiply the degree measure by pi over 180, and here's the little formula right here. Degree times pi over 180. So here's a sample problem worked out for you already. We are going to convert 76 degrees to radian measure. So we are going to multiply 76, the degree measure, by pi over 80. And when we do that, we get 76pi over 180, but then, of course, we always have to reduce the fraction. And both 76 and 180 are divisible by 4. And when we divide 76 by 4, we get 19. And when we divide 180 by 4, we get 45. So 76 degrees is equal to 19pi over 45.
So now let's try one from the beginning and see how it goes. So we're gonna convert 275 degrees into radian measure. So we know we have to do 275 times pi over 180. Well, that just gives us 275pi divided by 180. But of course, you got to reduce that fraction. So clearly, both 275 and 180 are divisible by 5. So let's bring up the calculator and see what happens: 275 divided by 5 equals 55. And 180 divided by 5 equals 36. Fifty-five and 36 have no numbers that can divide into both of those, so we know we've got our answer, which is going to be 55pi divided by 36. And there's our answer. So 275 degrees is equal to 55pi over 36 radians.
Now let's look about going the other way. So this might seem pretty obvious to you to convert radian measures into degree measures, multiply the radian measure by 180 over Pi. So again, that means we're gonna multiply the radian times 180 over Pi, and you can see the formula right here. And again, we've got the first example already worked out: 7pi over 8 radians. We want to convert that into degrees. So we're gonna do 7pi over 8 times 180 over pi, just using this formula. Of course, when you have pi over pi, the pi's are gonna cancel out. And then when we do 7 times 180 divided by 8, we wind up with 157.5 degrees.
So now let's try this one. We're gonna convert 4.2 radians into degrees. So we know that means that I have to do 4.2 radian measure times 180 divided by pi. So now we're gonna get our calculator and see what happens. So we're gonna do 4.2 times 180, and that equals 756, but then we have to divide that by pi. And we get 240.64 about. So I'll round this to 240.64. So we have 240.64 degrees. So it's really easy to convert back and forth between radians and degrees and degrees and radians.
Let's look at the calculator for just a sec. The first thing you need to know when you're dealing with radian measures in the calculator is you have to be sure that your calculator's in the right mode. So we're gonna click on the mode button, which is just right up here on the top. And you will see that the third one down we can choose either to be in radian or degree measure. So right now we're gonna go down and we're gonna highlight radian and we're gonna click inner. So now we're in radian mode. So now we can operate in radian mode.
So you can also see we have right here sine, cosine, and tangent. So if we want to find the sine of 3pi over 4, we just type that in. And remember the pi button is up above the little carat button and we have to hit the blue button to get there, as you can see from the key orders there. Now I'm gonna hit enter and we get .7071067812. And let's pull up just real quickly our unit circle. And we did the sine of 3pi over 4, and that says it's supposed to be the square root of 2 over 2. Well, let's see what happens in the calculator when we do that. So we're gonna go second square root of 2, which equals that, and then we divide that by 2. And lo and behold, we get exactly the same thing. So we have verified that our unit circle is giving us the right answers.
Suppose I want to do tangent of 7pi divided by 6. So I just typed it in and I get .5773502692. And you will recall the tangent is sine over cosine. So let's go to our unit circle again and we looked at 7pi over 6. And tangent was sine over cosine, so we need to divide negative one-half by negative square root of 3 over 2. So let's see what happens. Negative one-half divided by negative 3 raised to the .5 divided by 2. And we get exactly the same number.
So again, we're seeing that we can use either the calculator or our unit circle to get the answers that we're looking for.
Now here's a slightly different kind of problem. If cosine of X equals 0.234, find X. Well, X is inside the cosine, which creates a problem. And we've got to find a way to release that. And the way we're going to release that is by taking the arc sine, or the inverse sine, of both sides. So we're gonna do arc cosine of cosine of X equals the arc cosine of 0.234. Now, please also note that the arc cosine equals the cosine raised to the negative first power of X. Both of those are the same. However, we need to know that cosine negative one to the X – the cosine raised to the negative first of X is not equal to one over cosine of X. This is not true 'cause if this were true, then the inverse cosine would be equal to secant 'cause one over cosine is equal to the secant. So this relationship does not work.
So simply know that cosine raised to the negative 1 of X is the same thing as the inverse cosine. It's the same thing as the arc cosine of X. So once we take this, we know that inverse functions cancel each other out. So the arc cosine and the cosine are gonna cancel each other out and leave behind X equals arc cosine of 0.234. Now, to finish solving this, we're gonna have to use the calculator. So I'll bring the calculator up. We'll clear it out so we can start from the beginning. And you will notice up above cosine we have cosine raised to the negative first power. And we just talked about the fact that arc cosine is equal to cosine to the negative first power, so we can use this to get where we need to go. So we're gonna go second and hit the cosine button, and notice that puts cosine to the negative first power, which is the inverse cosine or arc cosine.
And we have 0.234 that we're trying to find. So I just type that in and we'll hit enter and we'll get 1.334606442. And again, we'll round this to two places, so we'll say it's approximately 1.33. So 1.33. So in this case, X is about 1.33. And we are in radians, so this would be radians. We could also find it in degrees. To do that, we would have to come back into our calculator, hit the mode button, come down here, highlight degree, and now we're back in degree mode. And now we can do second cosine and get the inverse cosine up there: 0.234, hit enter, and we have 76.467-something degrees, but we'll round this to 76.47. So this is also equal to about 76.47 degrees.
So here's a similar problem. If sine of X equals 0.52, find X. Well, again, just like we did with the arc cosine, we're gonna take the arc cosine of sine X or we can do sine raised to the negative first power of the sine of X equals the sine raised to the first power of 0.52. And again, just like before, inverse sine and sine cancel each other out, leaving the X behind, so we have X equals inverse sine of 0.52. So now we'll go back to our calculator. Our calculator's in degree mode, so we'll start with degrees and we'll go second sine to bring up the inverse sine. And we'll then type in the 0.52 and close that out and hit enter and we get 31.33 about. So this is equal to about 31.33 degrees. And now we'll go with finding the radian.
So we'll come into mode, we'll come down and highlight radian, hit enter. Now we're back in radian mode. So now we can go second sine to get the inverse sine, .52, hit enter, and we have about .546, so we'll round that to .55. So we have .55 radians. Another one—this time we'll look at tangent. If tangent X equals 2.3, find X. Well, we can do our tangent or tangent raised to the first power of both sides. It doesn't matter. They're both the same thing. So we'll do arc tangent of tangent of X equals arc tangent of 2.3. And again, just like you've been seeing, the arc tangent and tangent cancel out and we are left with X equals the arc tangent of 2.3. And we'll come back to our calculator. We're in radian mode, and this time we'll just do radians.
There's tangent, and above that you will see that we have the inverse tangent. So I'll hit second tangent, and then we'll type in 2.3 and close it out and hit enter, and we get 1.16 about. So this is equal to 1.16 radians. Now, you may be wondering, "What if I want to do a problem like this: If cotangent of X equals 1.5, find X?" Well, on our calculator, cotangent doesn't exist, but we do know that cotangent is the reciprocal of tangent. So we can do it this way. We can do one divided by second tangent, and then we can type in our 1.5 and hit enter. And there we have our answer, 1.017 and so on. So we'll round this to 1.02. So in this case, X equals 1.02 radians. And we can do the same thing if we want to deal with secant or cosecant.
You'll remember that secant equals one divided by cosine and cosecant equals one divided by sine. So by the same measure, secant negative one of X will equal one divided by cosine to the negative one of X. And cosecant to the negative one of X will equal one divided by sine raised to the negative one of X. And we just saw that cotangent to the negative one of X is equal to one divided by the tangent raised to the negative one of X. So we can take advantage of these reciprocal relationships to find X in situations where we're looking at secant, cosecant, or cotangent simply by using 1 divided by 10 to the negative 1 or cosine to the negative 1 or sine to the negative 1, depending on what it is we want to find.
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