The Six Trigonometric Functions

Properties Domain and Range

Cotangent Properties
Range- (-∞, ∞)
Domain- All real numbers except where tan(θ) = 0
Period-π

Tangent Properties
Range- (-∞,∞)
Domain- All real numbers except where cos(θ) = 0
Period-π

Cosine Properties
Range- [-1,1]
Domain-All real numbers
Period-2π

Cosecant Properties
Range- (-∞, -1] ∪ [1, ∞)
Domain- All real numbers except where sin(θ) = 0
Period-2π

Sine Properties
Range- [-1,1]
Domain- All real numbers
Period-2π

Periodic Functions- Any mathematical function that repeats its values over a predetermined period of time or range of values is said to be periodic.
The smallest value is called period.

Secant Properties
Range- (-∞, -1] ∪ [1, ∞)
Domain- All real numbers except where cos(θ) = 0
Period-2π

Functions

COSINE

Given a unit cricle COSINE function
cos= x value

ADJACENT/HYPOTENUSE

TANGENT

Given a unit cricle TAN function
cos= y/x X cant = 0

OPPOSITE/ADJACENT

COTANGENT

1/TANGENT

Given a unit cricle COT function
cot= x/y Y cant =0

SINE

OPPOSITE/HYPOTENUSE

Given a unit cricle SINE function
sin = y value

SECANT

1/COSINE

Given a unit cricle SEC function
csc=1/x X cant =0

COSECANT

Given a unit cricle CSC function
csc= 1/y Y can't = 0

1/SINE

What are Trigonometric Functions?
Trigonometric Functions also known as circular functions are mathematical functions of an angle. They are used to connect the ratios of the triangle's sides to the angles of a right triangle.

Identities- An identity is an equation which for no matter what value you put in for your input there going to be equal.

Reciprocal Identitity

Quotient Identity

Pythagorean Identity

sin^2(θ)+cos^2(θ)=1
Customary notation

1+cot^2θ=csc^2θ

(sin θ)^2+ (cosθ)^2=1
Commutative Property of Addition

tan^2θ+1=sec^2θ

(cosθ)^2+(sinθ)^2=1
Substitute trigonometric function values for x and y

Even Odd Properties-
Even properties state that a function is even if f(-θ)=f(θ)
and f is odd if f(-θ)= -f(θ)

Sin(-θ)
ODD

Cos(-θ)
EVEN

Tan(-θ)
ODD

Cot(-θ)
ODD

Sec(-θ)
EVEN

Csc(-θ)
ODD

SOHCAHTOA-is the term used
to remember the relationship
between the trigonometric functions
(Tangent,Sine,Cosine) as well as sides of
a right triangle.

CAH
Cosθ=
ADJACENT
_____________
HYPOTENUSE

TOA
Tanθ=
OPPOSITE
________________
ADJACENT

SOH
sinθ=
OPPOSITE
______________
HYPOTENUSE

Inverse Fucntions

Inverses of the six trigonometric are formally known as
inverse or arc trigonometric functions. These functions are used to find angles based on the ratio of sides of a right triangle. Also they are used to solve equations.

Sin^-1
This inverse function gives the given angle whose sine
is equal to a given value.
Domain -1 ≤ x ≤ 1
Range -π/2 ≤ θ ≤ π/2

Cos^-1
This inverse function gives the given angle whose cosine
is equal to a given value.
Domain -1 ≤ x ≤ 1
Range 0 ≤ θ ≤ π

Tan^-1
This inverse function gives the given angle whose tangent
is equal to a given value.
Domain All real numbers
Range -π/2 < θ < π/2

Sec^-1
This inverse function gives the given angle whose sine
is equal to a given value.
Domain x ≤ -1 or x ≥ 1
Range 0 ≤ θ ≤ π or π < θ ≤ 2π

Csc^-1
This inverse function gives the given angle whose cosecant
is equal to a given value.
Domain x ≤ -1 or x ≥ 1
Range -π/2 ≤ θ ≤ π/2 excluding 0

Cot^-1
This inverse function gives the given angle whose cotangent
is equal to a given value.
Domain All real numbers
Range 0 < θ < π