Trigonometric Functions

sin (θ) is

opposite/hypotenuse && y-coordinate/radius

cos (θ) is

adjacent/hypotenuse && x-coordinate/radius

tan (θ) is

opposite/adjacent && y-coordinate/x-coordinate

cot (θ) is

adjacent/opposite && 1/[tan (θ)] && x-coordinate/y-coordinate

csc (θ) is

hypotenuse/opposite && 1/[sin (θ)] && radius/y-coordinate

sec (θ) is

hypotenuse/adjacent && 1/[cos (θ)] && radius/x-coordinate

sin (-θ)

-sin (θ)

cos (-θ)

cos (θ)

sin (30)=cos (60)

1/2

sin (45)=cos (45)

√2/2

sin (60)=cos (30)

√3/2

cos (30)=sin (60)

√3/2

cos (45)=sin (45)

√2/2

cos (60)=sin (30)

1/2

tan (30)

√3/3

tan (45)

1

tan (60)

√3

cof: sin θ=

cos (90-θ)

cof: cos θ=

sin (90-θ)

cof: tan θ=

cot (90-θ)

cof: cot θ=

tan (90-θ)

cof: sec θ=

csc (90-θ)

cof: csc θ=

sec (90-θ)

coterminal

if two or more angles have the same terminal (ending) side, the angles are said to be coterminal.
To do this, add or subtract multiples of 360

positive rotation

counter clockwise

negative rotation

clockwise

Unit circle

Quadrants:
1(Positive: all, negative: none)
2(Positive: sin, negative: cos, tan)
3(Positive: tan, negative: sin, cos)
4(Positive: cos, negative: sin, tan)

sin (θ)=

90=1
180=0
270=-1
360=0

cos (θ)=

90=0
180=-1
270=0
360=1

tan (θ)=

90=undefined
180=0
270=undefined
360=0

Radian measure equation

θ=s/r

Basic circular functions

the arc lengths on the unit circle is the same as the radian measure of the angle θ

sin s =

second coordinate=y

cos s=

first coordinate=x

tan s=

second coordinate/first coordinate=y/x (x≠0)

csc s=

1/second coordinate=1/y (y≠0)

sec s=

1/first coordinate=1/x (x≠0)

cot s=

first coordinate/second coordinate=1/y (y≠0)

tan (-x)=

-tan x

(sin s)^2+(cos s)^2=

1

(csc s)^2-(cot s)^2=

1

(sec s)^2-(tan s)^2=

1

sin θ=

cos (pi/2-θ)

cos (u+v)=

(cos u)(cos v)-(sin u)(sin v)

cos (u-v)=

(cos u)(cos v)+(sin u)(sin v)

cos α=

sin (pi/2-α)

sin (u-v)=

(sin u)(cos v)-(cos u)(sin v)

sin (u+v)=

(sin u)(cos v)+(cos 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)]

csc (-θ)=

-csc (θ)

cosecant graph

Period: 2pi
Domain: All real numbers except k(pi), where k is an integer
Range: (-∞, -1] U [1, ∞)

sine graph

Period: 2pi
Domain: All real numbers
Range: [-1, 1]
Amplitude: 1
Odd: sin(-s)=-sin s (goes from left to right up) origin: (0,0)

cosine graph

Period: 2pi
Domain: All real numbers
Range: [-1, 1]
Amplitude: 1
Even: cos(-s)=cos s Origin: (0,1)

tangent graph

Period: pi
Domain: All real numbers except (pi/2)+k(pi)
Range: All real numbers (right(down) to left (up))

cotangent graph

Period: pi
Domain: All real numbers except k(pi), where k is an integer
Range: All real numbers (right(up) to left (down))

secant graph

Period: 2pi
Domain: All real numbers except (pi/2)+k(pi), where k is an integer
Range: (-∞, -1] U [-1, ∞)

negative angle identities

sin(-X) = - sinX , odd function

csc(-X) = - cscX , odd function

cos(-X) = cosX , even function

sec(-X) = secX , even function

tan(-X) = - tanX , odd function

cot(-X) = - cotX , odd function

Double Angle identities

sin(2x)=2 sin x cos x
cos(2x)=(cos x)(cos x)-(sin x)(sin x)
tan(2x)=(2 tan x)/(1-tan^2 x)

Half Angle identities

sin x/2=±sqrt[(1-cos x)/2]
cos x/2=±sqrt[(1+cos x)/2]
tan x/2=±sqrt[(1-cos x)/(1+cos x)]

Inverse sine function

y=sin^-1 x
Domain: [-1,1] Range: [-pi/2, pi/2]

inverse cosine function

y=cos^-1 x
Domain: [-1,1] Range: [0,pi]

inverse tan function

y=tan^-1 x
Domain: (-∞,∞) Range: [-pi/2, pi]

sine compositions

sin(sin^-1 x)=x
sin^-1(sin x)=x

cosine compositions

cos(cos^-1 x)=x
cos^-1(cos x)=x

tan compositions

tan(tan^-1 x)=x
tan^-1(tan x)=x

Law of sines

a/sin(A)=b/sin(B)=c/sin(C)

Law of cosines

a^2=b^2+c^2-2bc(cos(A))
b^2=a^2+c^2-2ac(cos(B))
c^2=a^2+b^2-2ab(cos(C))

SSS, SAS

Law of cosines

ASA, AAS, SSA

Law of sines

AAA

Nothing, unsolvable

Absolute value of Complex Number

|a+bi|=sqrt(a^2+b^2)=r

Trigonometric Notation for Complex Numbers

a+bi=r(cos x+ i sin x)

sin x

b/r, b=r sin x

cos x

a/r, a=r cos x

Complex Numbers: Multiplication Trig Notation

r(cos x1 + i sin x1)*r2(cos x2+ i sin x2)
= r*r2(cos (x1+x2)+ i sin (x1+x2))

Complex Numbers: Division Trig Notation

r(cos x1+i sin x1)/r2(cos x2 + i sin x2)=
r/r2[cos (x1-x2)+ i sin(x1-x2)]

Demoivre's Theorem

For any complex number r(cos x + i sin x) and any natural number n,
[r(cos x + i sinx)]^n= r^n(cos nx+ i sin nx)

Roots of Complex Numbers

The nth roots of a complex numbers r(cos x+ i sin x), r !=0, are given by
r^(1/n)[cos (x/n+ k*360/n) + i sin (x/n+ k*360/n)]
where k=0, 1, 2, ....., n-1

convert polar coordinates

r=sqrt(x^2+y^2)
cos θ= x/r x=r cos θ
sin θ= y/r y=r sin θ
tan θ=y/x

convert equations

substitute r cos θ and r sin θ for x and y
and sqrt(x^2+y^2) for r

x-4+4y=y^2

Only one variable is squared, so this cannot be a circle, an ellipse or a hyperbola. Find an equivalent equation: x=(y-2)^2
This is the equation of a parabola

3x^2+3y^2=75

both variables are squared, so this cannot be a parabola. The squared terms are added, so this cannot be a hyperbola. Divide by 3 on bothsides to find an equivalent equation.
x^2+y^2=25, this is the equation of a circle.

y^2=16-4x^2

Both variables are squared so this cannot be a parabola. Add 4x^2 on bothsides to find an equivalent equation: 4x^2+y^2=16. The squared terms are added, so this cannot be a hyperbola. The coefficients of x^2 and y^2 are not the same, so this is not a circle. Divide by 16 on both side to find an equivalent equation.
x^2/4+y^2/16=1
Equation of an ellipse

x^2=4y^2+36

Both variables are squared, so this cannot be a parabola. Subtract 4y^2 on both sides to find an equivalent equation: x^2-4y^2=36. The squared terms are not added so this cannot be a circle or ellipse. Divide by 36 on both sides to find an equivalent equation.
x^2/36-y^2/9=1
This is the equation of a hyperbola.

rotation of axes (x)

x=x' cos θ- y'sin θ
x'=x cos θ+y sin θ
y=x' sin θ+y' cos θ
y'=-x sin θ+ y cos θ