For more information, please see full course syllabus of Trigonometry
For more information, please see full course syllabus of Trigonometry
Related Articles:
 Navigating the High School Math Maze
 Unit Circle: How to Memorize & Use
 Unit Circle Chart & Blank Practice Chart (PDF)
Sine and Cosine Values of Special Angles
Main definitions and formulas:
 A 454590 triangle has side lengths in proportion to 11√ 2.
 A 306090 triangle has side lengths in proportion to 1√ 32.

Degrees Radians Cosine Sine 0 0 1 0 30 π 6√ 3 21 245 π 4√ 2 2√ 2 260 π 31 2√ 3 290 π 20 1  Use these values to find sines and cosines in other quadrants. The mnemonic ASTC (All Students Take Calculus) helps you remember which ones are positive in which quadrant. (All, Sine, Tangent, Cosine)
Example 1:
Convert 120^{° } to radians, identify its quadrant, and find its cosine and sine.Example 2:
Convert (5π/3)^{R} to degrees, identify its quadrant, and find its cosine and sine.Example 3:
Identify all angles between 0 and 2π whose sine is − [1/2], in both degrees and radians, and identify which quadrant each is in.Example 4:
Convert 225^{° } to radians, identify its quadrant, and find its cosine and sine.Example 5:
Identify all angles between 0 and 2π whose cosine is − (√3/2), in both degrees and radians, and identify which quadrant each is in.Sine and Cosine Values of Special Angles
 First convert 315^{°} to radians by using the following formula: degree measure ×[(π)/(180^{°} )] = radian measure
 315^{°} ×[(π)/(180^{°} )] = [(315^{°}π)/(180^{°} )] = [(7π)/4]
 Locate [(7π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
 Looking at our picture, we see that [(7π)/4] is in quadrant IV. This will help us to solve for the sine and cosine values as well.
 Note that 315^{°} is 45^{°} from the x  axis so we can use the 45^{°} − 45^{°} − 90^{°} triangle to find the sine and cosine.
 Recall that cosine is the x coordinate and sine is the y coordinate of [(7π)/4] on the unit circle
 cos [(7π)/4] = [(√2 )/2] and sin [(7π)/4] = − [(√2 )/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since [(7π)/4] is in quadrant IV we know that cosine will be positive and sine will be negative
 First convert [(4π)/3] to degrees by using the following formula: radian measure ×[(180^{°})/(π)] = degree measure
 [(4π)/3] ×[(180^{°})/(π)] = 4 ×60^{°} = 240^{°}
 Locate 240^{°} on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
 Looking at our picture, we see that 240^{°} is in quadrant III. This will help us to solve for the sine and cosine values as well.
 Note that [(4π)/3] is [(π)/3] past the x  axis so we can use the 30^{°} − 60^{°} − 90^{°} triangle to find the sine and cosine.
 Recall that cosine is the x coordinate and sine is the y coordinate of 240^{°} on the unit circle
 cos 240^{°} = − [1/2] and sin 240^{°} = − [(√3 )/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since 240^{°} is in quadrant III we know that tangent will be positive so sine and cosine will be negative
 First convert 210^{°} to radians by using the following formula: degree measure ×[(π)/(180^{°})] = radian measure
 210^{°} ×[(π)/(180^{°})] = [(210^{°}π)/(180^{°})] = [(7π)/6]
 Locate [(7π)/6] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
 Looking at our picture, we see that [(7π)/6] is in quadrant III. This will help us to solve for the sine and cosine values as well.
 Note that 210^{°} is 30^{°} past the x  axis so we can use the 30^{°} − 60^{°} − 90^{°} triangle to find the sine and cosine.
 Recall that cosine is the x coordinate and sine is the y coordinate of [(7π)/6] on the unit circle
 cos [(7π)/6] = − [(√3 )/2] and sin [(7π)/6] = − [1/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since [(7π)/6] is in quadrant III we know that tangent will be positive so cosine and sine will be negative
 First convert [(3π)/4] to degrees by using the following formula: radian measure ×[(180^{°})/(π)] = degree measure
 [(3π)/4] ×[(180^{°})/(π)] = 3 ×45^{°} = 135^{°}
 Locate 135^{°} on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
 Looking at our picture, we see that 135^{°} is in quadrant II. This will help us to solve for the sine and cosine values as well.
 Note that [(3π)/4] is [(π)/4] from the x  axis so we can use the 45^{°} − 45^{°} − 90° triangle to find the sine and cosine.
 Recall that cosine is the x coordinate and sine is the y coordinate of 135^{°} on the unit circle
 cos 135^{°} = − [(√2 )/2] and sin 135^{°} = [(√2 )/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since 135^{°} is in quadrant II we know that sine will be positive and cosine will be negative
 First convert 150^{°} to radians by using the following formula: degree measure ×[(π)/(180^{°})] = radian measure
 150^{°} ×[(π)/(180^{°})] = [(150^{°}π)/(180^{°})] = [(5π)/6]
 Locate [(5π)/6] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine and cosine values.
 Looking at our picture, we see that [(5π)/6] is in quadrant II. This will help us to solve for the sine and cosine values as well.
 Note that 150^{°} is 30^{°} from the x  axis so we can use the 30^{°}  60^{°}  90^{°} triangle to find the sine and cosine.
 Recall that cosine is the x coordinate and sine is the y coordinate of [(5π)/6] on the unit circle
 cos [(5π)/6] = − [(√3 )/2] and sin [(5π)/6] = [1/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since [(5π)/6] is in quadrant II we know that sine will be positive and cosine will be negative
 Locate [(3π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine value.
 Looking at our picture, we see that [(3π)/4] is in quadrant II. This will help us to solve for the sine value.
 Note that [(3π)/4] is [(π)/4] from the x  axis so we can use the 45^{°} − 45^{°} − 90^{°} triangle to find the sine value.
 Recall that sine is the y coordinate of [(3π)/4] on the unit circle
 sin [(3π)/4] = [(√2 )/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since [(3π)/4] is in quadrant II we know that sine will be positive.
 Locate 330^{°} on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the cosine value.
 Looking at our picture, we see that 330^{°} is in quadrant IV. This will help us to solve for the cosine value.
 Note that 330^{°} is 30^{°} from the x  axis so we can use the 30^{°}  60^{°}  90^{°} triangle to find the cosine value.
 Recall that cosine is the x coordinate of 330^{°} on the unit circle
 cos 330^{°} = [(√3 )/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since 330^{°} is in quadrant IV we know that cosine will be positive.
 Locate [(5π)/4] on a unit circle (draw a picture). This will help you to determine the quadrant it is in and it will also help you to find the sine value.
 Looking at our picture, we see that [(5π)/4] is in quadrant III. This will help us to solve for the sine value.
 Note that [(5π)/4] is [(π)/4] past the x  axis so we can use the 45^{°} − 45^{°} − 90^{°} triangle to find the sine value.
 Recall that sine is the y coordinate of [(5π)/4] on the unit circle
 sin [(5π)/4] = − [(√2 )/2]
 Using the Mnemonic ASTC (All Students Take Calculus) helps to determine which values are positive in which quadrant
 Since [(5π)/4] is in quadrant III we know that sine will be negative.
 Recall that cosine is the x coordinate.
 Draw a unit circle and locate positive [1/2] along the x  axis. Draw a vertical line through that point. This will help you to identify the two points on the circle.
 Connect the points on the circle to the origin. Notice that a 30^{°}  60^{°}  90^{°} triangle was formed.
 The first angle is 60^{°} past 0^{°} ⇒ 0^{°} + 60^{°} = 60^{°}
 The second angle is 30^{°} past of 270^{°} ⇒ 270^{°} + 30^{°} = 300^{°}
Angle 1  Angle 2  
Degree  60^{°}  300^{°} 
Radian  [(π)/3]  [(5π)/3] 
Quadrant  I  IV 
 Recall that sine is the y coordinate.
 Draw a unit circle and locate − [(√3 )/2] along the y  axis. Draw a horizontal line through that point. This will help you to identify the two points on the circle.
 Connect the points on the circle to the origin. Notice that a 30^{°}  60^{°}  90^{°} triangle was formed.
 The first angle is 60^{°} past 180^{°} ⇒ 180^{°} + 60^{°} = 240^{°}
 The second angle is 60^{°} short of 360^{°} ⇒ 360^{°} − 60^{°} = 300^{°}
Angle 1  Angle 2  
Degree  240^{°}  300^{°} 
Radian  [(4π)/3]  [(5π)/3] 
Quadrant  I  IV 
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Sine and Cosine Values of Special Angles
Lecture Slides are screencaptured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
 Intro 0:00
 454590 Triangle and 306090 Triangle 0:08
 454590 Triangle
 306090 Triangle
 Mnemonic: All Students Take Calculus (ASTC) 5:21
 Using the Unit Circle
 New Angles
 Other Quadrants
 Mnemonic: All Students Take Calculus
 Example 1: Convert, Quadrant, Sine/Cosine 13:11
 Example 2: Convert, Quadrant, Sine/Cosine 16:48
 Example 3: All Angles and Quadrants 20:21
 Extra Example 1: Convert, Quadrant, Sine/Cosine
 Extra Example 2: All Angles and Quadrants
Trigonometry Online Course
I. Trigonometric Functions  

Angles  39:05  
Sine and Cosine Functions  43:16  
Sine and Cosine Values of Special Angles  33:05  
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D  52:03  
Tangent and Cotangent Functions  36:04  
Secant and Cosecant Functions  27:18  
Inverse Trigonometric Functions  32:58  
Computations of Inverse Trigonometric Functions  31:08  
II. Trigonometric Identities  
Pythagorean Identity  19:11  
Identity Tan(squared)x+1=Sec(squared)x  23:16  
Addition and Subtraction Formulas  52:52  
Double Angle Formulas  29:05  
HalfAngle Formulas  43:55  
III. Applications of Trigonometry  
Trigonometry in Right Angles  25:43  
Law of Sines  56:40  
Law of Cosines  49:05  
Finding the Area of a Triangle  27:37  
Word Problems and Applications of Trigonometry  34:25  
Vectors  46:42  
IV. Complex Numbers and Polar Coordinates  
Polar Coordinates  1:07:35  
Complex Numbers  35:59  
Polar Form of Complex Numbers  40:43  
DeMoivre's Theorem  57:37 
Transcription: Sine and Cosine Values of Special Angles
Hi we are working out some examples of common values of sin and cos of common angles.0000
Remember what we learned, everything comes back to knowing those two key triangles.0007
There is the 45, 45, 90 triangle whose values are (root2)/2, (root 2)/2 and 1.0014
And then there is the 30, 60, 90 triangle whose values are 1/2, (root 3)/2 and 1.0025
If you remember those set of numbers, you can work out sin and cos of any common value anywhere on the unit circle.0038
That is what we are doing here, we have given the values 225 degrees, convert it to radians, let us start with that.0046
225 x pi/180 is equal to, well 225/180 simplifies down to 5/4, so that is 5pi/4.0053
Let us draw that on the unit circle and see where it lands.0069
My unit circle is 0, pi/2, which is the same as 90 degrees, pi radians is equal to 180 degrees, and 3pi/2 radians is 270 degrees, 2pi radians is equal to 360 degrees.0085
We have got 225 degrees or 5pi/4, 225 is between 180 and 270, in fact it is exactly half way between there because it is 45 degrees from either side.0105
It is right there, if you like that in terms of radians, 5pi/4 is just pi + pi/4, that is the angle we are looking at.0118
That is in the third quadrant, we found its quadrant, we converted it to radians, we want to find its cos and sin, that is the x and y coordinates.0133
Let me draw those in there, we want to figure out what those x and y coordinates are.0146
Look that is a common triangle and I remember what the values of those common triangles are.0150
That distance is (root 2 )/2, that distance is (root 2)/2.0157
Iâ€™m getting that because I remember this common 45, 45, 90 right triangle.0162
That means the sin and cos of both (root 2)/2 and we just need to figure out whether they are positive or negative.0171
All students take calculus, down there on the third quadrant only the tan is positive, both the sin and cos are negative.0188
Another way to remember that is just to remember that in the third quadrant both x and y values are negative.0201
Sin and cos of this angle are both negative (root 2)/2.0207
Again, what we are using over and over in these examples is these two key triangles.0220
The 45, 45, 90 triangle and the 30, 60, 90 triangle, you want to remember the values for those two triangles, (root 2)/2, (root 3)/2, and Â½.0228
Remember those values and then when you have an angle that is in the other quadrants, it is just a matter of translating one of those triangles over there.0244
And then figuring out whether the sin and cos are positive and negative.0252
Let us try one more example here, we want to identify all the angles between 0 and 2 pi whose cos is â€“root 3/2.0000
I start by drawing my unit circle that is not quite straight, let me straighten that up a little bit.0010
Cos is â€“ root 3/2, now the cos remember is the (x) value so Iâ€™m going to go on the (x) axis and Iâ€™m going to go to â€“root 3/2.0037
There it is.0053
That is â€“root 3/2 on the (x) axis and then Iâ€™m going to draw and see what angles I will get from that.0058
It looks like, remember that root 3/2 is one of my common values, that means that the y values are going to be Â½.0073
I need to figure out which angles those are but that is one of my common values Â½ root 3/2 that means that is a 30 degree angle, that is 60 and that is 30.0083
I just need to figure out what those angles are, if you remember we started 0, 90, 180, 270, and 360.0097
That first angle there is 30 degrees short of 180, the first angle is 150 degrees.0111
The second angle is 30 degrees past 180, so that is 210 degrees.0120
I have got my angles in degrees I will convert them into radians x pi/180 is equal to 5pi/6 to 10 x pi/180 is 7pi/6 radians.0127
I got those two angles in radians now, that is the first one 5pi/6, that is the second one 7pi/6.0151
And identify which quadrant each one is in, one of them is in the second quadrant, one of them is in the third quadrant, quadrant 2 and quadrant 3.0159
It all comes back to recognizing those common values, Â½, square root of 3/2, square root of 2/2.0184
Once you recognize those common values, you can put these triangles in any position anywhere on the unit circle.0191
You just figure out where is your root 3/2, where is your Â½, where is your root 2/2 and then you figure out which one is positive and which one is negative.0198
The whole point of this is you can figure out the sin and cos of any angle anywhere on the unit circle as long as it is a multiple of 30 or 45, or in terms of radians if it is a multiple of pi/6, pi/6, pi/4, pi/3.0209
You can figure out sin and cos of all these angles just by going back to those 3 common values and by figuring out whether their sin and cos are positive or negative.0226
Now you know how to find sin and cos of special angles, this is www.educator.com, thanks for watching.0238
Hi, these are the trigonometry lectures on educator.com.0000
Today we're going to learn about sine and cosine values of special angles.0003
When I say these special angles, there are certain angles that you really want to know by heart.0009
Those are the 454590 triangle, and the 306090 triangle.0014
Let me talk about the 454590 triangle first.0022
I'll draw this in blue.0028
Here's a 454590 triangle and I'm going to say that each side has length 1.0041
If each of the short sides has length 1, by the Pythagorean theorem, we can figure out that the long side, the hypotenuse, would have length square root of 2.0052
I'm going to scale this triangle down a little bit now.0066
I wanted to scale it down so the hypotenuse has length 1.0071
That means I have to divide all three sides by square root of 2.0075
If I scale this down, so the hypotenuse has length 1 that means the shorter sides has length 1 over the square root of 2, because I divided each side by square root of 2.0079
Then if you rationalize that, the way you learned in your algebra class, multiply top and bottom by square root of 2. 0095
You get square root of 2 over 2, and square root of 2 over 2.0102
Those are very important values to remember because those are going to come up as sines and cosines of our 45degree angles on the next slide.0113
First, I'd like to look also at the 306090 triangle.0122
I have to do a little geometry to work this out for you.0129
I'm going to start with an equilateral triangle, a triangle where all three sides have 60 degrees.0133
I'm going to have assume that each side has length 2.0142
The reason I'm going to do that is because I'm going to divide that triangle in half.0145
If we divide that triangle in half, then we get a right angle here and each one of these pieces will have length 1.0150
Now, if I just look at the right hand triangle.0160
Remember that each one of the corners of the original triangle was 60 degrees.0169
That means that the small corner is 30 degrees, and I have a right angle here.0175
Now, the short side has length 1, the long side has length 2.0182
I'm going to figure out what the other side is using the Pythagorean theorem.0186
Let me call that x for now.0191
I know that x^{2} + 1^{2} = 2^{2}, which is 4.0194
So, x^{2} = 4  1, which is 3.0202
So, x is the square root of 3.0209
That's where I got this relationship, 1, square root of 3, 2.0213
There's the 1, there's the square root of 3, and there's the 2.0219
Now, I'm going to turn this triangle on its side, and I want to scale it down.0223
Originally, it was 1, square root of 3, 2.0232
But again, I want to scale the triangle down so that the hypotenuse has length 1.0238
To do that, I have to divide everything by 2.0244
So the short side now has length 1/2.0247
The longer of the two short sides has length square root of 3 over 2.0251
Remember that's the side adjacent to the 30degree angle. 0258
That's the side adjacent to the 60degree angle, 0262
That's the right hand side.0264
These two triangles are very key to remember, in remembering all the sines and cosines.0267
In fact, if you can remember these lengths of these two triangles, you can work out everything else just from these two triangles.0273
Let me emphasize again, the 454590 triangle, its sides have length 1, square root of 2 over 2, and square root of 2 over 2.0282
The 306090 triangle, has sides of length 1, 1/2, and root 3 over 2.0297
Those are the values that you need to remember.0310
If you can remember those, you can work out all the sines and cosines you need to know for every trigonometry class ever.0312
Let's explore those a little bit.0321
We already figured out, let me draw a unit circle.0324
We know that sines and cosines occur as the x and y coordinates of different angles. 0328
Well, if you're at 0... 0338
Let me just draw in some key angles here, 0, here's 90, here's 45, here's 30, and here's 60.0341
If you're at 0 degrees, which is the same as 0 radians, then the cosine and sine, the x and y coordinates are just 1 and 0.0359
We already figured those out before.0368
The other easy one is the 90degree angle up here.0370
We figured out that that's π/2 radians, and the cosine and sine are 0, and 1 there.0374
Now, the new ones, let me start with 45, because I think that one's a little bit easier.0380
The 45degree angle, there it is right there.0386
We want to figure out what the x and y coordinates are because those give us the sine and cosine.0391
Well, we just figured out that a triangle that has 1 as its hypotenuse has square root of 2 over 2, as both its x and y sides.0397
That's where we get the square root of 2 over 2 as the cosine and sine of the 45degree angle, also known as π/4 radians.0407
For the 30degree angle, I'll do this one in blue.0418
The 30degree angle, we have again, hypotenuse has length 1.0422
Remember, the length of the long side is root 3 over 2.0431
And the length of the short side is 1/2.0436
That's how you know the sine and cosine of the 30degree angle or π/6.0439
The cosine, the xcoordinate, root 3 over 2, sine is 1/2.0447
The 60degree angle, that's just the same triangle but it's flipped the other way so that the long side is on the vertical part and the short side is on the horizontal axis.0452
The short side is 1/2.0468
The long side is now the yaxis, that's root 3 over 2.0473
That's how we get 1/2 being the cosine of 60 degrees, root 3 over 2 being the sine of 60 degrees.0478
These values are really worth memorizing but you remember that you figure all out from those two triangles.0488
All you need to know is that one triangle has length 1, has hypotenuse 1, and sides root 2 over 2, that's the 454590 triangle.0494
The other triangle has hypotenuse 1, and then the long side is root 3 over 2, short side is 1/2, that's the 306090 triangle.0509
Just take that triangle and you flip it whichever way you need to, to get the angle that you're looking for.0526
These angles, these sines and cosines are the key ones to remember, root 2 over 2, root 3 over 2, and 1/2.0534
From that, what you're going to do is figure out the sines and cosines of all the other angles all over the unit circle.0546
Here's my unit circle.0573
We figured out all the sines and cosines of all the angles in the first quadrant.0575
All we have to do now is figure out all the sines and cosines of the angles in the other quadrants.0583
Let me draw them out.0589
But it's the same numbers every time.0590
All you have to do is figure out whether those numbers are positive or negative, and that just depends on which quadrant you're in.0593
All you have to do is remember those key numbers, root 2 over 2, root 3 over 2 and 1/2.0601
Then you're going to figure out which ones are positive in which quadrants.0607
Let me show you how you'll remember that, 1, 2, 3, 4.0613
Remember that the sine is the y value, and the cosine is the x value.0622
In the first quadrant, both the x's and y's are positive, and so sines and cosines are going to positive.0630
I'm going to write that as an a, which stands for all the values of everything is positive.0636
In the second quadrant, over here, the x values are negative, the y values are positive.0643
Now, the x values correspond to the cosines, the cosines are negative and the sines are positive.0653
I'm going to write as s here, for the sines being positive.0658
In the third quadrant, both x's and y's are negative.0667
X and y are both negative, that means cosines and sines are both negative.0669
Tangent, we haven't learned the details of tangent yet but we're going to learn later that tangent is sine over cosine.0675
Both sine and cosine are negative, that means sine over cosine is positive.0683
It turns out that tangent is positive down that third quadrant.0687
We'll learn the details of tangent later.0691
In the meantime, we'll just remember that tangent is positive in the third quadrant.0693
In the fourth quadrant here, that was the third quadrant that we just talked about, now we're moving on to the fourth quadrant.0698
The x's are positive now, the y's are negative.0705
That means the cosine is positive, but the sine is negative.0708
I'll list the cosine here because I'm listing the positive ones.0713
First quadrant, they're all positive.0718
Second quadrant, sines are positive.0720
Third quadrant, tangents are positive.0722
Fourth quadrant, cosines are positive.0724
The way you remember that is with this little acronym, All Students Take Calculus.0727
That shows you as you go around the four quadrants, All Students Take Calculus.0733
It shows you which ones are positive in each quadrant.0741
First quadrant are all positive.0744
Second quadrant sines are positive.0746
Third quadrant, tangents are positive.0748
Fourth quadrant, cosines are positive.0750
That's how you'll remember what the signs are in each quadrant.0753
That means the positive signs and the negative signs in each quadrant.0756
The numbers are just all these values that you've just memorized, root 3 over 2, root 2 over 2, and 1/2.0758
We'll do some practice finding sines and cosines of values in other quadrants, based on this table that we remember.0768
And these common values in common triangles that we remember.0775
Then we'll take those values, introduce some positive and negative signs, and we'll come up with the sines and cosines of angles in other quadrants.0780
Let's try some examples.0787
First example here is a 120 degrees, you want to convert that to radians, identify it's quadrant, and find it's cosine and sine.0791
First things first, let's convert it to radians, 120 times π/180 is 2π/3, because 120/180 is 2/3 so that's 2π/3 radians.0800
Identify it's quadrant.0806
Well, let me graph out my unit circle here.0814
There's 0, there's π/2, π, 3π/2, and then 2π.0820
Now, 2π/3 is between π/2 and π.0840
In fact, it's closer to π/2.0845
It's 2/3 of the way around from 0 to π.0848
If you like that in terms of degrees, π is 180 degrees, and so 2π/3 is 2/3 of the way over to 180.0852
Now, we're going to find cosine and sine.0862
Let me show you how to do this.0867
You draw a triangle here. 0868
Remember, we're looking for the x and y coordinates.0870
Draw a triangle here.0873
That's a 3060 triangle.0875
This is 60, that's a 60degree angle.0877
That's a 30degree angle.0887
We know what the lengths of these different sides are.0891
We know that the long side there is root 3 over 2 and the short side there is 1/2.0897
We know, remember, the cosines and sine are the x and y coordinates.0907
The cosine of 120 or 2π/3 is 1/2, except that we're going to have to check whether that's positive or negative.0910
Remember All Students Take Calculus.0930
In the second quadrant, only the sine is positive, so the cosine must be negative.0935
The sine of 2π/3, the y value is root 3 over 2.0942
In the second quadrant, sines are positive, so that's positive.0950
Our cosine and sine are 1/2 and root 3 over 2.0954
If you didn't remember the All Students Take Calculus thing, you can also just work it out once you know what quadrant it's in.0958
It's in quadrant 2 and we know there that the x coordinates are negative, and the y coordinates are positive.0964
The cosine must be negative and the sine must be positive.0974
The whole point of this is that you only really need to memorize the values of the triangles, root 2 over 2, root 3 over 2 and 1/2.0981
Once you know those basic triangles, you can work out what the sines and cosines are in any different quadrant just by drawing in those triangles and then figuring out which ones have to be positive, and which ones are to be negative.0991
Let's try another one.1005
This one is converting 5π/3 radians to degrees, identifying it's quadrant, and finding its cosine and sine.1009
5π/3 times 180/π, the π's cancel, and the we have 5/3 of 180, 180 over 3 is 80, so this is 5 times 60 is 300 degrees.1018
Let's try and find that in the unit circle.1040
We have 0, π/2 which is 90, π which is 180, 3π/2 which is 270, and 2π which is the same as 360 degrees.1053
Now, 5π/3, that's bigger than π and that's smaller than 2π.1074
In fact, that's π + 2π/3.1080
That's π, which is right here, plus 2π/3.1090
So, 2/3 the way around from π to 2π.1099
There it is right there.1103
That's in the fourth quadrant.1104
So, we figured out what quadrant it's in.1108
If you like degrees better, 300 degrees is a little bigger than 270, in fact, it's 30 degrees past 270, and 60 degrees short of 360.1110
That's how you know that that angle is in that quadrant.1121
Now we have to find its cosine and sine.1126
That's the x and y coordinates.1128
We set up our triangle there.1134
We already know that that's a 60degree angle, because it's 60 degrees short of 360.1137
That's a 30degree angle and we just remember our common values.1142
The horizontal value, that's the short one, that's 1/2, that's the long one, root 3 over 2.1147
We know our values are going to be 1/2 and root 3 over 2, we'll just have to figure out which one's positive and which one's negative.1154
I know my cosine of 5π/3 is going to be either positive or negative 1/2.1162
The sine of 5π/3 is positive or negative root 3/2.1171
Remember All Students Take Calculus.1179
Down there in the fourth quadrant, the cosine is positive and the sine is negative.1184
If you don't remember All Students Take Calculus, you just look that you're in the fourth quadrant, x coordinates are positive, y coordinates are negative.1189
You know which one's positive or negative.1199
All you have to remember are those key values, 1/2, root 3 over 2, root 2 over 2.1202
Remember those key values for the key triangles.1209
Then it's just a matter of drawing the right triangle in the right place and figuring out which one is positive, and which one is negative.1212
Let's try another one.1219
This one is kind of tricky.1223
This one's going to be challenging us to go backwards from the sine.1224
We have to find all angles between 0 and 2π whose sine is 1/2.1229
This is kind of foreshadowing the arc sine function that we'll be studying later on and one of the later lectures.1242
In the meantime, the sine is 1/2.1249
Remember now, the sine is the ycoordinate.1252
We want things whose y coordinates are 1/2.1256
I'm going to draw 1/2 on the yaxis, 1/2.1259
I'm going to look for all angles whose ycoordinate is 1/2.1266
Look, there's one right there.1272
And there's one right there.1276
I'm going to draw those in.1277
I'm going to draw those triangles in.1280
I know now that if we have a vertical component of 1/2, the horizontal component has to be root 3 over 2.1289
That's because we remember those common triangles, 1/2, root 3 over 2, root 2 over 2.1298
We're going to figure out what those angles are.1307
I know that's a 30degree angle.1312
I know that that is 180.1318
The whole thing is 210 degrees.1322
I know that that is 30.1330
I know that that must be 60.1334
Remember, this is 270 degrees down here.1337
We have 270 degrees plus 60 degrees.1342
This is getting a little messy, so I'm going to redraw it over here.1345
That's the angle we're trying to chase down here.1357
We know that's 60.1360
That much is 270.1362
So, 270 plus 60 is 330 degrees.1366
Those are the two angles that we're after, 210 degrees, 330 degrees.1372
Let me convert those into radians.1378
If you multiply that by π/180 then that's equal to...1382
Let's see, that's 7π/6 radians.1390
This one times π/180 is equal to 11π/6 radians.1397
We've got our two angles in degrees and radians.1409
The quadrants, the first one was in the third quadrant, quadrant 3.1413
The second one was in the fourth quadrant, quadrant 4.1420
Those are the two angles in both degrees and radians that had a sign of 1/2.1426
Their y value was 1/2.1436
What this comes down to is knowing those common values, 1/2 root 3 over 2, root 2 over 2.1442
Once you know those common values, it's a matter of looking at the different quadrants and figuring out whether the x and y values are positive or negative.1454
In this case, we had the sine, sine remember is the ycoordinate.1461
Since it was negative, we knew that we had to be at 1/2 on the yaxis.1468
We found 1/2 on the yaxis, drew in the triangles, recognized the 3060 triangles that we've been practicing and then we were able to work out the angles.1473
We'll try some more examples of that later.1484
1 answer
Last reply by: Dr. William Murray
Fri Jan 8, 2016 1:23 PM
Post by sania sarwar on January 7 at 03:04:12 AM
Hi Sir
thanks for the lectures they are really helpful but can you please suggest me on which lecture to watch for this problem because I cant get my head around it.
If sin(theta)=0.3,cos(x)=0.7 and tan (alpha)=0.4 then find sin(3pi/2+theta)
1 answer
Last reply by: Dr. William Murray
Thu Feb 19, 2015 5:27 PM
Post by patrick guerin on February 19, 2015
Why is it that when you type in the cosine or sine of a number on a calculator, you get something like .0679 or something?
3 answers
Last reply by: Dr. William Murray
Mon Nov 17, 2014 8:12 PM
Post by Tami Cummins on September 20, 2013
I know this is so simple but I really do not understand how 5pi/3 equals pi + 2pi/3.
1 answer
Last reply by: Dr. William Murray
Fri Jun 21, 2013 6:22 PM
Post by A C on June 17, 2013
Please help: why, if I use a calculator to find the angle value of sin/cos/tan, does sin and tan give me the angle (neg. or pos. depending on the quadrant obviously) but cos gives me are larger angle (>90degrees), what I assume is the standard position angle?
1 answer
Last reply by: Dr. William Murray
Wed May 22, 2013 3:25 PM
Post by Monis Mirza on May 17, 2013
hi,
In the extra example I, how did you know that the special triangle is
454590 triangle?
Thanks
1 answer
Last reply by: Dr. William Murray
Thu Nov 15, 2012 6:21 PM
Post by peter chrysanthopoulos on November 14, 2012
nevermind I got it
1 answer
Last reply by: Dr. William Murray
Thu Nov 15, 2012 6:20 PM
Post by peter chrysanthopoulos on November 14, 2012
how did you know that it was a 30/60/90 triangle in example 1?
1 answer
Last reply by: Dr. William Murray
Thu Apr 18, 2013 11:34 AM
Post by Dr. William Murray on October 17, 2012
Hi Chin,
It depends on which direction you're going. If it's radians to degrees, multiply by 180/pi. If it's degrees to radians, multiply by pi/180.
This makes sense if you follow the rules that you learn in physics and chemistry about units: 180 degrees = pi radians, so (180 degrees)/(pi radians) = 1. Then when you want to convert in either direction, you multiply by 1:
(3 pi/4 radians) x (180 degrees)/(pi radians) = 135 degrees.
90 degrees x (pi radians)/(180 degrees) = pi/2 radians.
Hope this helps. Thanks for studying trigonometry!
Will Murray
1 answer
Last reply by: Dr. William Murray
Thu Apr 18, 2013 11:35 AM
Post by chin chang on October 15, 2012
On the example problems you converted the degrees or radians into one or the other by multiplying pi/180 or 180/pi. How do you know what to multiply for each situation? Does it make sense? For instance on the second example problem, how did you to multiply 5pi/3 radians by 180/pi?
3 answers
Last reply by: Dr. William Murray
Fri Sep 28, 2012 4:54 PM
Post by Ivon Nieto Ivon Nieto on September 25, 2012
Is there an easier way to memorize the unit circle sine/cosine?