Post by: cosine on April 23, 2015, 07:05:38 am
The Unit Circle:
What is it?
The unit circle is a circle plotted on a Cartesian Plane. It has a centre (0, 0) and a radius of one. What does that mean though? It means that (check the image underneath) if you place a point anywhere on the circumference of the circle, the length from the origin to that point will always be one unit. It's equation is
(http://i.gyazo.com/d0f013697ae0d17da2c4cb3e820f7cda.png)
In the image above, we can see that the centre is put on a graph where x-axis and y-axis cross, so we have this neat arrangement. There are four sections named quadrants. We start from the positive x-axis, which is also labelled, and rotate in an anticlockwise direction from there. Angles also play an important role in trigonometry, which essentially define our quadrants.
Angles
What are they?
Every angle that we will deal with in trigonometry is measured from the positive x-axis. This feature is very important, and once you can get your head around it, the whole angle process will become easy to you. If we move in an anticlockwise direction from the positive x-axis, our angle will be a positive one. However, if we move in a clockwise direction from the positive x-axis, our angle will be negative. For simple explanations, I will be using degrees instead of radians. It is easier and more effective to understand the degree version, then only you can see how the radian angles work, so:
(http://i.gyazo.com/a0118dde7a95abf38bb0919de13035e7.png) (http://i.gyazo.com/9f330a5b7d51a70e7b9265694008caf0.png)
In the first image, because the angle starts from the positive x-axis, and projects in an anticlockwise direction, it is a positive angle. Similarly, the second picture displays a negative angle, as it projects in a clockwise (downwards) direction from the positive x-axis.
This is the way the angles are defined, whenever you see a negative angle, you know it moves clockwise from the positive x-axis, and if you are presented with a positive angle, you should know it moves in an anti clockwise direction.
Quadrants
What are they?
If we start from the positive x-axis, and move 90 degrees (note: positive angle, hence anti clockwise direction) from the x-axis, we come in contact with the y-axis. This portion is known as quadrant 1 where all the angles contained within are obviously between
(http://i.gyazo.com/c5c0c5b2ace76c6b1a2aa7b640bc7123.png)
If we continue moving another 90 degrees, we come in contact with the negative x-axis. So, from our original position, (positive x-axis) we have moved a total of 90+90= 180 degrees. Quadrant 2 is defined between 90 and 180 degrees, where all angles in this quadrant are between
(http://i.gyazo.com/ca33da702d80ce5747bede330b56c769.png)
Again, move another 90 degrees, now at 270. Between 180 and 270 is known as quadrant 3, where all angles are between
(http://i.gyazo.com/c432c2f12305fe6a8fd84cd985b2b870.png)
Finally, another 90 degrees and we have made a full anti clockwise rotation of 360 degrees. The portion between 270 to 360 degrees is known as quadrant 4, where all angles are between
(http://i.gyazo.com/c1f9eb3f4197e5091e824cf48be1c1d7.png)
Recap:
Quadrant 1:
Quadrant 2:
Quadrant 3:
Quadrant 4:
Examples:
(http://i.gyazo.com/fe6f3fefe016b7a5dbe0895d1a2d4af3.png)
(http://i.gyazo.com/5b9c3d4dbd6b09b56887ce7dcfdf7185.png)
Sine
What is it?
We define
(http://i.gyazo.com/e2cb52690692f639294da7729dea8605.png)
Cosine
What is it?
We define
(http://i.gyazo.com/4ed654713b6961dd6468f86355d691d4.png)
Tangent
What is it?
The tangent function can be though of as the line x=1, where the ray OP is extended to the point T. Tangent is only positive in quadrants 1 and 3, and negative in quadrants 2 and 4.
(http://i.gyazo.com/6f5dd92c9fd1e532934e39526f7ba786.png)
Recap:
~ Y-coordinate made at the angle
~ Positive where y-axis is positive (quadrants 1, 2)
~ Negative where y-axis is negative (quadrants 3, 4)
~ X-coordinate made at the angle
~ Positive where the x-axis is positive (quadrants 1, 4)
~ Negative where the x-axis is negative (quadrants 2, 3)
~ The tangent made through x=1 at the angle
~ Positive in quadrants 1, 3
~ Negative in quadrants 2, 4
Some helpful links:
https://www.mathsisfun.com/geometry/unit-circle.html
https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/Trig-unit-circle/v/unit-circle-definition-of-trig-functions-1
http://www.intmath.com/blog/mathematics/unit-circle-an-introduction-5166
http://samples.jbpub.com/9781449606046/06046_CH03_123-178.pdf
http://online.math.uh.edu/MiddleSchool/Modules/Module_4_Geometry_Spatial/Content/UnitCircleTrigonometry-TEXT.pdf
https://www.youtube.com/watch?v=cIVpemcoAlY
https://www.youtube.com/watch?v=YK7KoU6ELWg
https://www.youtube.com/watch?v=j5SoWzBSUmY
I hope you understood what was said above, if you have any problems, suggestions or questions please do not hesitate to PM me, or leave a comment on this thread! Thank you!
Part 2: How NOT to Memorise Exact Values! - Trigonometry
Post by: TheAspiringDoc on April 23, 2015, 12:58:31 pm
Cheers :)
Post by: keltingmeith on April 23, 2015, 01:40:15 pm
Can anyone explain to me why 1 radian equals tan(89)?
Cheers :)
It doesn't?
Post by: cosine on April 23, 2015, 05:04:55 pm
Can anyone explain to me why 1 radian equals tan(89)?
Cheers :)
What makes you think it equals 1, show me your working out and Ill be able to help :)
Post by: TheAspiringDoc on April 23, 2015, 05:54:34 pm
What makes you think it equals 1, show me your working out and Ill be able to help :)No, I was thinking that 1 Radian equals Tan (89)..
I know they show up slightly differently on my calculator (by like .1) but maybe it is just a coincidence?
:D
Post by: keltingmeith on April 23, 2015, 06:03:17 pm
No, I was thinking that 1 Radian equals Tan (89)..
I know they show up slightly differently on my calculator (by like .1) but maybe it is just a coincidence?
:D
1 radian=1 normal number, though. That's why we use radians. So, tan(89) only equals 1 radian if tan(89)=1, but it doesn't. (depending on where 89 is in degrees or radians, it's actually 1.6 or like 54 or something else fairly massive)
Post by: TheAspiringDoc on April 23, 2015, 06:17:06 pm
1 Radian = 57.295779 degrees...
Post by: cosine on April 23, 2015, 06:38:42 pm
Tan (89 degrees) = 57.28996...
1 Radian = 57.295779 degrees...
As mentioned above, sin, cos and tan are only lengths. So when you say tan(89)=57, that's a length. But when you say 1 Radian = 57 degrees, that's completely different!
Loving the curiosity though, definitely bound to do well in VCE :D
Post by: AngelWings on April 23, 2015, 10:27:01 pm
For those of you who can't seem to recall whether sine or cosine are associated with which axis, go back to pre-VCE maths and recall the acronym SOH CAH TOA (for those confused/ going "I've never heard of it in my life", see attachment). As these are right-angled triangles (SOH CAH TOA can only be used in this instance), you may apply it to the unit circle.
The unit circle, as Cosine has stated, has a radius of one unit.
For sine and cosine, note that the radius is actually the hypotenuse. (If you don't believe me, check the diagrams in Cosine's post.) Therefore, radius = hypotenuse = 1 unit and substitute into SOH CAH TOA.
For sine, you are left with the "opposite" side and for cosine, you are left with the "adjacent" side.
When you see a diagram, you will notice it will be parallel to the y-axis in the case of sine (see red arrows on sine diagram in Cosine's post). Moreover, it will be on the x-axis in the case of cosine (see red arrows on the cosine diagram in Cosine's post).
Post by: cosine on April 23, 2015, 10:33:26 pm
Suggestion here.
For those of you who can't seem to recall whether sine or cosine are associated with which axis, go back to pre-VCE maths and recall the acronym SOH CAH TOA (for those confused/ going "I've never heard of it in my life", see attachment). As these are right-angled triangles (SOH CAH TOA can only be used in this instance), you may apply it to the unit circle.
The unit circle, as Cosine has stated, has a radius of one unit.
For sine and cosine, note that the radius is actually the hypotenuse. (If you don't believe me, check the diagrams in Cosine's post.) Therefore, radius = hypotenuse = 1 unit and substitute into SOH CAH TOA.
For sine, you are left with the "opposite" side and for cosine, you are left with the "adjacent" side.
When you see a diagram, you will notice it will be parallel to the y-axis in the case of sine (see red arrows on sine diagram in Cosine's post). Moreover, it will be on the x-axis in the case of cosine (see red arrows on the cosine diagram in Cosine's post).
Well explained! Thank you for the addition of the info! :)
Post by: kinslayer on April 23, 2015, 10:46:50 pm
What is the definition of tan(x) for quadrants 2, 3 and 4?
Thanks!
Same definition in all quadrants. tan is positive in quadrants 1 and 3, negative in 2 and 4.
Post by: AngelWings on April 23, 2015, 10:51:32 pm
Well explained! Thank you for the addition of the info! :)
No problems. I will be back to explain common memorising techniques for the exact values during the weekend. I think that a lot of people hesitate to recall those numbers off the top of their head. Maybe you should have a go first, Cosine.
Post by: cosine on April 24, 2015, 07:18:51 am
No problems. I will be back to explain common memorising techniques for the exact values during the weekend. I think that a lot of people hesitate to recall those numbers off the top of their head. Maybe you should have a go first, Cosine.
Sure, will work on it tonight! :D
Post by: 99.90 pls on April 24, 2015, 07:30:53 am
Same definition in all quadrants. tan is positive in quadrants 1 and 3, negative in 2 and 4.
I meant visually. The definition provided here is "The tangent function can be though of as the line x=1, where the ray OP is extended to the point T. Tangent is only positive in quadrants 1 and 3, and negative in quadrants 2 and 4."
Does that mean for Quadrant 3, the ray cuts the entire unit circle?
Post by: kinslayer on April 24, 2015, 12:57:09 pm
I meant visually. The definition provided here is "The tangent function can be though of as the line x=1, where the ray OP is extended to the point T. Tangent is only positive in quadrants 1 and 3, and negative in quadrants 2 and 4."
Does that mean for Quadrant 3, the ray cuts the entire unit circle?
For Quadrant 3, the tangent would be the length of that part of the line x = -1 between the x-axis and its intersection with the ray, like a mirror image of Quadrant 1.
For a definition of the tangent, you really only need to know that it is equal to sin/cos, but this way can be enlightening as well; for example, it shows intuitively why the tangents of angles close to pi/2 or -pi/2 should be so large and why the tangent is undefined at those points.
Post by: AngelWings on April 25, 2015, 11:16:48 pm
Memorising Tips for Exact Values
Choose whichever way works best for you. Please feel free to amend (I didn't check this very well.) or post your ways of memorising them, if you feel it is going to be beneficial.
Let's begin by reminding ourselves of the exact values.
Exact Values
They are often listed by resources in the form a table, which you can see right here.
Method 1: Memorising through Repetition and Practise
Spoiler
The most basic method is to just keep using them and do tonnes of questions requiring your exact values. Textbooks usually have quite a large collection of questions of this sort. Seeing it being used in different, but similar, contexts will often make it easier to recall.
Method 2: Exact Values Triangles
Spoiler
Some people are visual learners. This is one way that'll satisfy those people and usually favoured by teachers (or at least in my experience, this statement is true).
There are two triangles that you must memorise in order to take this route. People usually call them the "exact value triangles" for obvious reasons.
The 30o and 60o Triangle
The 45o Triangle
Alternatively, you can see the exact value triangles in pictorial version:30 and 60 degree triangle and 45 degree triangle.
As I have stated in my previous post (see page 1), we can only apply SOH CAH TOA to right-angled triangles. Since they are, you can easily apply these ratios on these triangles to receive your exact values.
A breakdown of the steps required:
There are two triangles that you must memorise in order to take this route. People usually call them the "exact value triangles" for obvious reasons.
The 30o and 60o Triangle
Spoiler
The first triangle is equilateral with sides of 2 units and, being equilateral, will have 180/3 = 60o angles. If we break the triangles in half, we get two equal right-angled triangles of sides 2 units (the hypotenuse), 1 unit (the bottom side, opposite the 30o) and the square root of 3 units(the left side, opposite the 60o).
Okay, let's pause. Here's a question for you: Where did these sides come from?
Okay, let's pause. Here's a question for you: Where did these sides come from?
- The 2 units side. SpoilerThe 2 units comes directly from the equilateral triangle. It is the length of one of the equilateral triangles' sides.
- The 1 unit side. SpoilerThis is the side that was halved in the equilateral triangle, so the length of the original equilateral triangle, which was 2 units, is now halved, i.e. 1 unit.
- The square root of 3 side: If you don't know where the square root of 3 comes from, you should try using Pythagoras' theorem (spoiler).SpoilerPythagoras's theorem can be applied only for right-angled triangles, which we have. The formula is: a2+b2=c2 (or h2, if you want to call it "h" for hypotenuse, but in this case, let's just go with c2). The hypotenuse is equivalent to the length of c. The pro-numerals "a" and "b" refer to the other two sides.
Substitute a=1 (see dot point number 2) and c=2 units (see dot point number 1).
You should receive: 12+b2=22. What we now want is to get "b" as the subject. So a little rearrangement gives: b2=4-1 and therefore b=square root of 3 units. That's how we got square root of 3 as the side.
Spoiler
This is a right-angled, isosceles triangle with sides 1 unit, 1 unit and the square root of 2 units (hypotenuse).
Again, question: Where does the hypotenuse's length come from?
Again, question: Where does the hypotenuse's length come from?
Spoiler
It can be worked from Pythagoras' theorem. Substitute a=1 unit and b=1 unit to find c's length (the hypotenuse).
As I have stated in my previous post (see page 1), we can only apply SOH CAH TOA to right-angled triangles. Since they are, you can easily apply these ratios on these triangles to receive your exact values.
A breakdown of the steps required:
Spoiler
- Note which triangle you must use and whether you want to work out sin, cos or tan.
- Substitute values for theta and the sides in the ratio.
Spoiler
1. Use the 60o triangle
2. apply tan(theta)=opposite/adjacent and substitute values.
3. receive tan(60o)= sqrt(3)/1= sqrt(3) units
2. apply tan(theta)=opposite/adjacent and substitute values.
3. receive tan(60o)= sqrt(3)/1= sqrt(3) units
Method 3a: Knowing Consecutive Numbers and Memorising Tangent Values Separately
Spoiler
This is the one that works for me. It's more difficult to prove, but it's kind of a weird pattern that occurs with sin and cos.
Somehow, when you memorise the table, you will find that all the sin values increase consecutively from 0 to 4 and then the square root is taken and finally divided by 2. That is:
It is the converse for cos values i.e. from 4 to 0, since they are complementary.
For tan values, you can memorise them separately.
Somehow, when you memorise the table, you will find that all the sin values increase consecutively from 0 to 4 and then the square root is taken and finally divided by 2. That is:
Spoiler
sin(0o)=sqrt(0)/2=0
sin(30o)=sqrt(1)/2=1/2
sin(45o)=sqrt(2)/2=1/sqrt(2) (rationalised)
sin(60o)=sqrt(3)/2
sin(90o)=sqrt(4)/2=2/2=1
sin(30o)=sqrt(1)/2=1/2
sin(45o)=sqrt(2)/2=1/sqrt(2) (rationalised)
sin(60o)=sqrt(3)/2
sin(90o)=sqrt(4)/2=2/2=1
For tan values, you can memorise them separately.
Method 3b: Knowing Consecutive Numbers and Knowing that Tangent Values Increase in Quadrant 1
Spoiler
This is essentially the same method as Method 3a.
If you see the graph of tan(x), you will note that from 0 to 90o, it increases from 0 to positive infinity. For tan values, it works with you memorising the exact values and then putting them in increasing order i.e. 0, 1/sqrt(3), 1, sqrt(3) and infinity. These will then pair up with the angles in increasing order respectively. In other words: tan(0o)=0, tan(30o)=1/sqrt(3), tan(45o)=1 and so forth.
If you see the graph of tan(x), you will note that from 0 to 90o, it increases from 0 to positive infinity. For tan values, it works with you memorising the exact values and then putting them in increasing order i.e. 0, 1/sqrt(3), 1, sqrt(3) and infinity. These will then pair up with the angles in increasing order respectively. In other words: tan(0o)=0, tan(30o)=1/sqrt(3), tan(45o)=1 and so forth.
EDIT 2: Method 4: The Finger or Hand Trick (Alchemy's suggestion)
Spoiler
As suggested by Alchemy, there is also the hand trick. I, personally, have only encountered this once and didn't remember it until it was mentioned. (Thanks Alchemy.) Yeah... so I'm obviously not the best person to explain it. Instead I decided to get you guys some resources from people who can.
Helpful resources to learn/ recall the hand trick:
Trigonometric Finger Trick PDF
Unit Circle Hand Trick
Simple Hand Trick for Memorizing Exact Values (A 5 and a half minute video for any AN visual learners out there.)
Helpful resources to learn/ recall the hand trick:
Trigonometric Finger Trick PDF
Unit Circle Hand Trick
Simple Hand Trick for Memorizing Exact Values (A 5 and a half minute video for any AN visual learners out there.)
Applying This to Other Quadrants
You can apply these exact values in any quadrant, as long as you can remember whether the value is positive or negative for that quadrant and if that angle's equivalent (axis of symmetry). Particularly tricky are the negative angles, because it makes you think in the opposite direction, but don't fret, a little practise will make you better at them.
Also, if you need to solve them in radians, it's the same, except you need to remember the conversion between degrees and radians.
Question: How do we convert...
a. from degrees to radians?
Spoiler
Multiply the amount of degrees by pi/180.
Spoiler
Multiply the amount of radians by 180/pi.
Note: I've never done extensive notes like these before on here, so I'm open to constructive criticism. I apologise for my lack of LaTeX and computer knowledge and used the most simple method to cater for the amount of time I had to write this.
Post by: cosine on April 25, 2015, 11:36:59 pm
Here is my How NOT to Memorise Exact Values! - Trigonometry, the above methods is absolutely easier to remember, but the way I show it may help you remember them visually. Hope you guys enjoy :D
Post by: AngelWings on April 25, 2015, 11:45:46 pm
Go AngelWings! Good job on the explanations!
Here is my How NOT to Memorise Exact Values! - Trigonometry, the above methods is absolutely easier to remember, but the way I show it may help you remember them visually. Hope you guys enjoy :D
Yeah, thanks Cosine. I would have gone visual, but time constraints.
Anyway, Cosine's done a great job explaining it visually.
(200 posts)
Post by: MathsNerd203 on April 26, 2015, 06:01:03 am
Post by: cosine on April 26, 2015, 07:27:14 am
Could you do one of these for calculus, probability and matrices?
Of course! I will be doing calculus next, thanks for the suggestion! :)
Post by: alchemy on April 26, 2015, 10:07:20 pm
might be useful up to the point when they become etched into your head anyway...
Post by: AngelWings on April 26, 2015, 11:04:54 pm
someone show 'em the hand trick as well for remembering exact values ;)
might be useful up to the point when they become etched into your head anyway...
Just edited post. Consider that done.
Post by: achre on April 27, 2015, 12:33:30 am
Post by: cosine on April 27, 2015, 07:10:22 am
What an apt username. Great work!
Haha yeah! Thanks man :D
Post by: Arithmetic on April 27, 2015, 07:17:18 pm
Post by: cosine on April 29, 2015, 06:33:00 pm
Great work cosine!!!
No worries buddy, glad it helped!
Post by: cosine on June 20, 2015, 02:21:11 pm
How to Sketch Circular Functions Easily!