PreCalculus

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Chapter 5: Trigonometric Function

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PreCalculus

Chapter 8

Area of a Triangle

A = 1/2(b * h)

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

Law of Sines

A + B + C = 180 degrees

sinA/a = sinB/b sinA/a = sinC/c sinB/b = sinC/c

sinA/a = sinB/b = sinC/c

Right Triangle

SOH-CAH-TOA

cotθ = 1/tanθ = b/a

secθ = 1/cosθ = c/b

csc = 1/sinθ = c/a

tanθ = a/b

cosθ = b/c

sinθ = a/c

Chapter 7

Product to Sum and Sum to Product

Sum to Product formulas

cos(a)- cos(b)= -2sin((a+b)/2)sin((a-b)/2)

cos(a)+ cos(b)= 2cos((a+b)/2)cos((a-b)/2)

sin(a) - sin(b) = 2sin((a-b)/2)cos((a+b)/2)

sin(a)+(b)sin = 2sin((a+b)/2)cos((a-b)/2)

Product to Sum formulas

sin(a)cos(b)= 1/2[sin(a+b) + sin (a-b)]

cos(a)cos(b) = 1/2[cos(a-b) + cos(a+b)]

sin(a)sin(b) = 1/2[cos(a-b) - cos(a+b)]

Double & Half Angles

Half-Angle Formula

tan(α/2) = sqrt((1-cos a)/(1+cos a))

cos(α/2) = sqrt((1+cos a)/2)

sin(α/2) = sqrt((1-cos a)/2)

Double-angle Formulas

tan^2θ = (1-cos(2θ))/(1+cos(2θ))

cos^2θ = (1+cos(2θ))/2

(sin^2)θ = (1-cos(2θ))/2

tan(2θ) = (2tanθ)/(1-tan^2θ)

cos(2θ) = 2(cos^2)θ-1

cos(2θ) = 1-2(sin^2)θ

cos(2θ) = (cos^2)θ-(sin^2)θ

sin(2θ) = 2sinθcosθ

Sum and Differences

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))

Formulas for Sine Function

sin (a-b) = sin(a)cos(b)- cos(a)sin(b)

sin (a+b) = sin(a)cos(a)+ cos(a)sin(b)

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)

Trig Identities

An trigonometric identity is defines as two functions are to be identical for every value of x on the functions.

Even-Odd Identites

cot(-θ) = -cotθ

tan(-θ) = -tanθ

sec(-θ) = secθ

cos(-θ) = cosθ

csc(-θ) = -cscθ

Sin(-θ) = -sinθ

Pythagorean Identities

tan^2θ+1=sec^θ

sin^2θ+cos2=1

Reciprocal Identities

cotθ=1/tanθ

secθ=1/cosθ

cscθ=1/sinθ

Quotient Identities

cotθ=cosθ/sinθ

tanθ=sinθ/cosθ

Inverse Trig Functions

If x=sin(y) then y=sin^-1(x), similar with all other functions.

Reflective around the orgin

Range of f = domain of f^-1

Domain of f = range of f^-1

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))

Chapter 6

Trig Graphs

Transformations

Horizonal Stretch/Compression

Vertical Stretch/Compression

Horizonal Shift

Vertical shift

Trig Functions:

Cotangent Function

Range = All Real Numbers

Domain = All real numbers except for (k*pi)

1/tan or also known as cos/sin

Secant Function

Domain = Any real numbers but pi/2 or 3pi/2

1/cos

Cosecant Function

Range = (-infinity,-1]U[1,infinity)

Domain = All real numbers except integer multiples of pi otherwise known as 180 degrees.

1/sin

Tangent Function

Range = All real Numbers

Domain = All real numbers except odd integer multiples of pi/2 otherwise know as 90 degrees

Association of t from the ratio of y-coordinate to the x-coordinate of P. Also known as sin/cos

Cosine Function

Associates angles of t with x-coordinates in the circumference of a unit circle

Sine

Range = [-1,1]

Domain = All real numbers

Associates angles of t with y-coordinates in the circumference of a unit circle.

Unit Circle

Can be used to very easily find the similarities between radians and degrees within the unit circle. To be exact.

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.

Angles

Subtopic

Area of a Sector Theorem

1/2(r^2)(theta)

Arc Length

The distance along the outside of a circle.

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

Revolutions

360 degrees is the entirety of a circle

Right angles = 90degrees

Rays