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#### cosine

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##### Introduction to Trigonometry!
« on: April 23, 2015, 07:05:38 am »
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So trigonometry is a major part of the Mathematical Methods 1/2 and 3/4 course, and I know for a fact that many people either struggle with it, or find it difficult to grasp the general gist of the concepts. I believe that the key to overcome this is to master the fundamentals to trig, so that you can manipulate the harder questions. I have decided to comprise a thread dedicated to the introduction of trigonometry, with my explanations of the concepts, including some really helpful links to interactive sites that can assist you in your understandings. I hope you can benefit from this, and if you think a friend can too, feel free to share it!

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 $x^2+y^2=1$. There are four areas within the unit circle named quadrants. We name them quadrants 1, 2, 3 and 4 in an anticlockwise direction from the positive x-axis.

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:

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.

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 $0 < \theta < 90$

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 $90 < \theta < 180$

Again, move another 90 degrees, now at 270. Between 180 and 270 is known as quadrant 3, where all angles are between $180 < \theta < 270$.

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 $270 < \theta < 360$

Recap:
Quadrant 1: $0<\theta<90$
Quadrant 2: $90<\theta<180$
Quadrant 3: $180<\theta<270$
Quadrant 4: $270<\theta<360$

Examples:

Sine
What is it?
We define  $\sin\theta$  as the y-coordinate of point P. This can be thought of as, in simpler terms, the y-coordinate made at the angle $\theta$. In the image below, the red arrows represent the y-value or y-coordinate that is made at the angle of theta. This y-value is defined as $\sin\theta$. From the properties of the y-axis, we know that the bottom portion of the cartesian plane (the y-axis under the x-axis, the two bottom quadrants: 3 and 4) that the y-axis is negative, hence the sin function is also negative there too. Therefore in the positive y-axis, in quadrants 1 and 2, sin is also positive.

Cosine
What is it?
We define  $cos\theta$  as the x-axis of the point P. This too can be thought of as the length of the x-value made at an angle of $\theta$. So, from the properties of the x-axis, we know that it is positive on the right side of the Cartesian plane, but negative in the left side of the plane. Therefore cos is also positive in quadrants 1 and 4, but negative in quadrants 2 and 3.

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.

Recap:

$sin\theta$:
~ Y-coordinate made at the angle $\theta$
~ Positive where y-axis is positive (quadrants 1, 2)
~ Negative where y-axis is negative (quadrants 3, 4)

$cos\theta$:
~ X-coordinate made at the angle $\theta$
~ Positive where the x-axis is positive (quadrants 1, 4)
~ Negative where the x-axis is negative (quadrants 2, 3)

$tan\theta$:
~ The tangent made through x=1 at the angle $\theta$
~ Positive in quadrants 1, 3
~ Negative in quadrants 2, 4

https://www.mathsisfun.com/geometry/unit-circle.html
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

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
« Last Edit: April 25, 2015, 11:32:45 pm by cosine »
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#### TheAspiringDoc

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##### Re: Introduction to Trigonometry!
« Reply #1 on: April 23, 2015, 12:58:31 pm »
0
Can anyone explain to me why 1 radian equals tan(89)?
Cheers

#### keltingmeith

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##### Re: Introduction to Trigonometry!
« Reply #2 on: April 23, 2015, 01:40:15 pm »
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Can anyone explain to me why 1 radian equals tan(89)?
Cheers

It doesn't?

#### cosine

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##### Re: Introduction to Trigonometry!
« Reply #3 on: April 23, 2015, 05:04:55 pm »
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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
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#### TheAspiringDoc

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##### Re: Introduction to Trigonometry!
« Reply #4 on: April 23, 2015, 05:54:34 pm »
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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?

#### keltingmeith

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##### Re: Introduction to Trigonometry!
« Reply #5 on: April 23, 2015, 06:03:17 pm »
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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?

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)

#### TheAspiringDoc

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##### Re: Introduction to Trigonometry!
« Reply #6 on: April 23, 2015, 06:17:06 pm »
0
Tan (89 degrees) = 57.28996...

#### cosine

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##### Re: Introduction to Trigonometry!
« Reply #7 on: April 23, 2015, 06:38:42 pm »
+1
Tan (89 degrees) = 57.28996...

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
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#### AngelWings

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##### Re: Introduction to Trigonometry!
« Reply #8 on: April 23, 2015, 10:27:01 pm »
+1
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).
« Last Edit: April 23, 2015, 10:30:34 pm by AngelWings »
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#### cosine

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##### Re: Introduction to Trigonometry!
« Reply #9 on: April 23, 2015, 10:33:26 pm »
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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!
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#### kinslayer

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##### Re: Introduction to Trigonometry!
« Reply #10 on: April 23, 2015, 10:46:50 pm »
+1
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.

#### AngelWings

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##### Re: Introduction to Trigonometry!
« Reply #11 on: April 23, 2015, 10:51:32 pm »
+1
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.
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#### cosine

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##### Re: Introduction to Trigonometry!
« Reply #12 on: April 24, 2015, 07:18:51 am »
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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!
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#### 99.90 pls

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##### Re: Introduction to Trigonometry!
« Reply #13 on: April 24, 2015, 07:30:53 am »
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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?
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#### kinslayer

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##### Re: Introduction to Trigonometry!
« Reply #14 on: April 24, 2015, 12:57:09 pm »
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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.
« Last Edit: April 24, 2015, 01:02:01 pm by kinslayer »