PreCalculus
Chapter 6
Angles
Rays
Right angles = 90degrees
360 degrees is the entirety of a circle
Revolutions
Radians
Also known as theta, radians are a new way to measure
angles within a circle. This is done by dividing the arc length by the radius of the circle. Otherwise known as S/r
Arc Length
The distance along the outside of a circle.
Area of a Sector Theorem
1/2(r^2)(theta)
Subtopic
Trig Functions:
Unit Circle
Unit Circle equation = (x^2) + (y^2) = 1
Since there is a radius of one, the unit circle is an
easy way to find and graph points on a circle.
Can be used to very easily find the similarities
between radians and degrees within the unit circle. To be exact.
Sine
Associates angles of t with y-coordinates in the circumference of a unit circle.
Domain = All real numbers
Range = [-1,1]
Cosine Function
Associates angles of t with x-coordinates
in the circumference of a unit circle
Domain = All real numbers
Range = [-1,1]
Tangent Function
Association of t from the ratio of y-coordinate
to the x-coordinate of P. Also known as sin/cos
Domain = All real numbers except odd integer
multiples of pi/2 otherwise know as 90 degrees
Range = All real Numbers
Cosecant Function
1/sin
Domain = All real numbers except integer
multiples of pi otherwise known as 180 degrees.
Range = (-infinity,-1]U[1,infinity)
Secant Function
1/cos
Domain = Any real numbers but pi/2 or 3pi/2
Range = (-infinity,-1]U[1,infinity)
Cotangent Function
1/tan or also known as cos/sin
Domain = All real numbers except for (k*pi)
Range = All Real Numbers
Trig Graphs
Transformations
Vertical shift
Horizonal Shift
Vertical Stretch/Compression
Horizonal Stretch/Compression
Chapter 7
Inverse Trig Functions
f^-1(f(x)) = x for every x in the domain of f and
f(f^-1(x)) = x for evey x in the domain of (f^-1))
Domain of f = range of f^-1
Range of f = domain of f^-1
Reflective around the orgin
If x=sin(y) then y=sin^-1(x), similar with
all other functions.
Trig Identities
An trigonometric identity is defines as two functions
are to be identical for every value of x on the functions.
Quotient Identities
tanθ=sinθ/cosθ
cotθ=cosθ/sinθ
Subtopic
Subtopic
Reciprocal Identities
cscθ=1/sinθ
secθ=1/cosθ
cotθ=1/tanθ
Pythagorean Identities
sin^2θ+cos2=1
tan^2θ+1=sec^θ
Even-Odd Identites
Sin(-θ) = -sinθ
csc(-θ) = -cscθ
cos(-θ) = cosθ
sec(-θ) = secθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
Sum and Differences
Formulas for Cosine Function
cos(a+b) = cos(a)cos(b) - sin(a) sin(b)
cos(a-b) = cos(a)cos(b)+ sin(a)sin(b)
Formulas for Sine Function
sin (a+b) = sin(a)cos(a)+ cos(a)sin(b)
sin (a-b) = sin(a)cos(b)- cos(a)sin(b)
Formulas for Tangent Function
tan(a+b) = (tan(a)+tan(b))/(1-tan(a)tan(b))
tan(a-b) = (tan(a)- tan(b))/(1+tan(a)tan(b))
Double & Half Angles
Double-angle Formulas
sin(2θ) = 2sinθcosθ
cos(2θ) = (cos^2)θ-(sin^2)θ
cos(2θ) = 1-2(sin^2)θ
cos(2θ) = 2(cos^2)θ-1
tan(2θ) = (2tanθ)/(1-tan^2θ)
(sin^2)θ = (1-cos(2θ))/2
cos^2θ = (1+cos(2θ))/2
tan^2θ = (1-cos(2θ))/(1+cos(2θ))
Half-Angle Formula
sin(α/2) = sqrt((1-cos a)/2)
cos(α/2) = sqrt((1+cos a)/2)
tan(α/2) = sqrt((1-cos a)/(1+cos a))
Product to Sum and Sum to Product
Product to Sum formulas
sin(a)sin(b) = 1/2[cos(a-b) - cos(a+b)]
cos(a)cos(b) = 1/2[cos(a-b) + cos(a+b)]
sin(a)cos(b)= 1/2[sin(a+b) + sin (a-b)]
Sum to Product formulas
sin(a)+(b)sin = 2sin((a+b)/2)cos((a-b)/2)
sin(a) - sin(b) = 2sin((a-b)/2)cos((a+b)/2)
cos(a)+ cos(b)= 2cos((a+b)/2)cos((a-b)/2)
cos(a)- cos(b)= -2sin((a+b)/2)sin((a-b)/2)
Chapter 8
Right Triangle
SOH-CAH-TOA
sinθ = a/c
cosθ = b/c
tanθ = a/b
csc = 1/sinθ = c/a
secθ = 1/cosθ = c/b
cotθ = 1/tanθ = b/a
Law of Sines
sinA/a = sinB/b = sinC/c
sinA/a = sinB/b sinA/a = sinC/c sinB/b = sinC/c
A + B + C = 180 degrees
Law of Cosines
c^2 = a^2 + b^2 - 2ab cos C
b^2 = a^2 + c^2 - 2ac cos B
a^2 = b^2 + c^2 - 2bc cos A
Area of a Triangle
A = 1/2(b * h)