Sum To Product
sin x + sin y=
sin x - sin y =
cos x + cos y =
cos x - cos y =
= 2 sin ( x+y /2) cos ( x-y /2)
=2 cos( x+y /2) sin( x-y /2)
=2 cos ( x+y /2) cos ( x-y /2)
= - 2 sin ( x+y /2) sin ( x-y /2)
100
Period for y = a sin bx
sin and cos = 2 pi/b
100
Degrees to Radians
degrees x (pi radians)/180
100
Sine Curve
starts at 0
goes up to 1 and down to -1
period = 2pi
100
Pythagorean Identities
sin sqrd + cos sqrd = 1
1 + tan sqrd = sec sqrd
1 + cot sqrd = csc sqrd
200
Sum and Difference Formula
sin & cos & tan
sin (u+v) = sin u cos v + cos u sin v
sin (u-v) = sin u cos v - cos u sin v
cos (u+v) = cos u cos v - sin u sin v
cos (u-v) = cos u cos v + sin u sin v
tan (u+v) = (tan u + tan v) / (1- tan u tan v)
tan (u-v) = ( tan u - tan v) / (1 + tan u tan v)
200
Amplitude
the absolute value of a in y = a sin bx
the tangents amplitude is undefined because it goes to + - infinity
200
Coterminal Angles
Angles that have the same initial sides and the same terminal sides
* to find it, take the the angle - 2pi
200
Cosine Curve
starts at (0,1) and goes down to -1
goes through pi/2 and 3pi/2
period = 2pi
200
Cofunction Identities
sin (pi/2 - u ) = cos u
cos ( pi/2 - u ) = sin u
tan ( pi/2 - u ) = cot u
cot ( pi/2 - u ) = tan u
sec ( pi/2 - u ) = csc u
csc ( pi/2 - u ) = sec u
300
Power Reducing Formuas
Sine squared
Cosine squared
Tangent squared
sin2 = (1- cos 2u) /2
cos2= (1 + cos 2u) /2
tan 2= (1-cos 2u)/(1+ cos 2u)
300
Even Function
for each x in the domain of the function
f(-x) = f(x)
cosine and secant are even functions and have y axis symmetry.
300
Reference angles
the acute angles formed by theta and the terminal side of the horizontal axis
300
Inverse Sine Curve (arc sin)
flip the x and y values
300
even/ odd identities
sin (-u) = -sin u
csc (-u)= - csc u
cos (-u)= cos u
ssec (-u) = sec u
tan (-u) =-tan u
cot (-u) = -cot u
400
Half Angle Formulas
Sin (u/2)
Cos & Tan
Sin (u/2) = +- root (1-cos u)/2
cos (u/2) = +- root (1 + cos u)/2
tan (u/2) = (1-cos u)/sin u = sin u / (1 + cos u)
400
Odd function
for each x in the domain of the function
f(-x) = -f(x)
sin, csc, and tan are odd functions and have origin symmetry
400
sin 30
sin 45
sin 60
pi/6 = 1/2
pi/4 = root 2/2
pi/3 = root 3/2
400
Tangent Curve
asymptotes
tangent goes through o and pi (intercepts) curves up left to right
asymptotes are at pi/2 and 3pi/2
to find the asymptotes do
bx-c = pi/2
bx-c = - pi/2
400
Product to Sum formulas
sin u sin v = (1/2)[ cos (u-v)-cos(u+v)]
cos u cos v= (1/2)[cos (u+v) + cos (u+v)]
sin u cos v = (1/2)[sin(u+v) + sin (u-v)]
cos u sin v = (1/2)[sin (u+v) - sin (u-v)]
500
Double Angle Formulas
sin 2u
cos 2u
tan 2u
sin 2u = 2 sin u cos u
cos 2u = cos sqrd u - sin sqrd u = 2 cos sqrd u -1 = 1- sin sqrd u
tan 2u= (2 tan u )/(1- tan sqrd u)
500
Damped Trig Functions
graph the y=x function and it's inverse
then graph the trig function to fit the two lines
500
r and s
r is the radius
s = 2 pi r
s/r = theta
500
Cotangent functions curve
through pi/2 and 3pi/2 (intercepts)
asymptotes at 0, pi, 2pi
curves down from left to right