Toby Opferman
toby@opferman.net
Triginometry
Welcome to trignometric functions and circles.
First let me introduce you the trignometric functions & Names.
cos(theta) = x/r = Cosine
sin(theta) = y/r = Sine
tan(theta) = y/x = Tangent
1/cos(theta) = sec(theta) = r/x = Secant
1/sin(theta) = csc(theta) = r/y = Cosecant
1/tan(theta) = cot(theta) = x/y = Cotangent
What the trignometric functions do:
Let's pretend this ugly shit is a real circle that intersects at
(1,0) (0,1) (-1,0) (0, -1) on the axies.
^
. . .
. | .
. | .
<---.---+-----.----->
. | .
... .
|
|
This is the unit circle.
What the trignometric functions do is give us information about a point
on the circle.
For instance.
Each
90
^
135 | 45
|
\|/
180 <------+-------> 0 (360)
/|\
|
225 | 315
|
270
Each direction from the origin has an angle. If you want to know the
location of a place on the circle you would use the Trig Functions.
For instance. If we have a circle of size r. To find the
X value at degree 0 we would cos(0) = x/r
Let us take the unit circle as above
cos(0) = 1/1 = 1
sin(0) = 0/1 = 0
Hence, (1, 0) is on the unit circle.
If your radius is not 1 then you would want to do the following:
cos(theta)*r = x
sin(theta)*r = y
If your radius is 20, then
cos(0)*20 = 1/1 * 20 = 20
Remeber, even though this works out, it always returns the value as if
it was radius 1. But, you can treat the R as it if was the R since it works out.
In theory it would be:
cos(0)*20 = 20/20 * 20 = 20
But it works out the same.
Now, there is another form of measurement called radians.
180 is half circle. PI is half circle in radians.
PI/2
^
3PI/4 | PI/4
|
\|/
PI <------+-------> 0 (2PI)
/|\
|
5PI/4 | 7PI/4
|
3PI/2
To convert from Degrees to Radians:
Angle*PI/180 = Radians
To Convert from Raidans to Degrees:
Radians*180/PI = Degrees
180 and PI are equivlent in the systems.
There are also inverse Trig Functions:
arccos or cos^-1
arcsin or sin^-1
arctan or tan^-1
Where:
cos^-1(x/r) = theta
sin^-1(y/r) = theta
tan^-1(y/x) = theta
So
tan^-1(tan(theta)) = theta
and
tan(tan^-1(y/x)) = y/x
At any point on the circle you can form a triangle.
/|
/ |
/ |
/ |
R / | Y
/ |
/ |
/Theta |
/_@______|
X
The X is how far it comes out along the X axis, the Y is how far up obviously.
R is a direct line from origin to (X,Y) at an angle.
If you read the vectors tutorial you could represent these as vectors:
Since adding two vectors that are on top of each other gets you the connecting
vector:
+ <0, Y> = sqrt(X^2 + Y^2) = R
R is the length.
R =
Always remeber this equation:
X^2 + Y^2 = R^2
and
R = Sqrt(X^2 + Y^2)
Also remeber that this is for circle at center 0,0
(X - h)^2 + (Y - k)^2 = R^2
For circles at center (h, k)
The reason that (X - h) is - and not + is because of this:
Center x = 1, X intercept = X = 5
0 1 5
---+---|-------*>
All trignometric functions work at the origin, so you have to
adjust the circle to be at the origin.
5 - 1 = 4. From 5 to 1 is a X of 4, which would be a radius of 4.
If the graph was at the origin the X component would be 4 not 5.
-1 0 5
-|--+----------*>
If Center was -1, then 5 - -1 = 5 + 1 = 6
That's a length of 6. You shift the graph over to be correct at the origin
so the trignometric functions can work correctly.
To go backwards, you can have negative angles.
-45 degrees = 315 degrees.
If you want to convert any angle to be from 0-360, you just
keep subtracting or adding 360 (2PI) until you get an angle/radian
in that range and it will be equivlent.
cos^2(theta) + sin^2(theta) = 1 at all times.
To do a simple proof of this we can show using a vector
You know a vector is a length and a direction.
r
Now, we want to find the radius.
Sqrt((r*cos(theta))^2 + (r*sin(theta))^2)
Sqrt(r^2*cos(theta)^2 + r^2*sin(theta)^2)
Notice now that r^2 is a common term.
Sqrt(r^2*(cos(theta)^2 + sin(theta)^2))
Now, notice that Radius = Length = r.
Notice there is an r^2 in there and that is a squareroot function.
And we know the answer is supposed to be r.
cos(theta)^2 + sin(theta)^2 would have to be 1 in order for this to work out
correctly. (theta must equal theta, they both must be the same)
Sqrt(r^2*1) = r
cos^2(0) + sin^2(0) = 1 + 0 = 1
cos^2(45) + sin^2(45) = .5 + .5 = 1
There are other properties of trignometric functions
sin(theta)
--------- = tan(theta)
cos(theta)
Some are easy to see like the above.
Here are some others:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x)
tan(2x) = 2tan(x)/(1 - tan^2(x))
sin^2(x) = (1-cos(2x))/2
cos^2(x) = (1+cos(2x))/2
tan^2(x) = (1-cos(2x))/(1+cos(2x))
sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2)
sin(x) - sin(y) = 2cos((x+y)/2)sin((x-y)/2)
cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2)
cos(x) - cos(y) = -2sin((x+y)/2)sin((x-y)/2)
sin(x)sin(y) = 1/2(cos(x-y) - cos(x+y))
cos(x)cos(y) = 1/2(cos(x-y) + cos(x+y))
sin(x)cos(y) = 1/2(sin(x+y) + sin(x-y))
cos(x)sin(y) = 1/2(sin(x+y) - sin(x-y))
1 + tan^2(x) = sec^2(x)
1 + cot^2(x) = csc^2(x)
sin(PI/2 - x) = cos(x)
csc(PI/2 - x) = sec(x)
sec(PI/2 - x) = csc(x)
cos(PI/2 - x) = sin(x)
tan(PI/2 - x) = cot(x)
cot(PI/2 - x) = tan(x)
sin(-x) = -sin(x)
csc(-x) = -csc(x)
sec(-x) = sec(x)
cos(-x) = cos(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
sin(x +/- y) = sin(x)cos(y) +/- cos(x)sin(y)
cos(x +/- y) = cos(x)cos(y) -/+ sin(x)sin(y)
tan(x +/- y) = (tan(x) +/- tan(y))/(1 -/+ tan(x)tan(y))
*NOTE* Look at the +/- order!!!! It's purposes put in those fashions.
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