## Unit Circle & Trig Functions

Toby Opferman http://www.opferman.net 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.