Trigonometry

Trigonometric Identities

Quotient

tanθ=sinθ/cosθ

cotθ=cosθ/sinθ

Just tanθ and cotθ are the only functions that hace Quotient Idebtities

Reciprocal

cscθ=1/sinθ

secθ=1/cosθ

cotθ=1/tanθ

Each primary function has a reciprocal identity

Pythagorean

sinθ^2+cosθ^2=1

secθ^2-tanθ^2=1

cscθ^2-cotθ^2=1

Each Pythagorean Identity is connected since each uses the prymary identity but some uses reciprocal too

Odd

sin⁡(-θ)=-sinθ

csc(-θ)=-cscθ

tan(-θ)=-tanθ

cot(-θ)=-cotθ

Odds with a negative angle results as negative

Even

cos(-θ)=cosθ

sec(-θ)=secθ

Evens with a negative angle results as positive

Sum/Difference

For cosine

cos⁡(a+b)=cos⁡a cos⁡b-sin⁡a sin⁡b

cos⁡(a-b)=cos⁡a cos⁡b+sin⁡a sin⁡b

For sine

sin⁡(a+b)=sin⁡a cos⁡b+cos⁡a sin⁡b

sin⁡(a-b)=sin⁡a cos⁡b-cos⁡a sin⁡b

For tangent

tan⁡(α+β)=(tan⁡α + tan⁡β)/(1-(tan⁡α tan⁡β) )

tan⁡(α-β)=(tan⁡α - tan⁡β)/(1+(tan⁡α tan⁡β) )

Double Angle

For sine

sin⁡(2θ)=2sinθcosθ

sin^2 θ=1-cos⁡(2θ)/2

For cosine

cos⁡(2θ)=cos^2 θ+sin^2 θ

cos⁡(2θ)=1-2sin^2 θ

cos⁡(2θ)=2cos^2 θ-1

cos^2 θ=(1+cos⁡(2θ))/2

For tangent

tan⁡(2θ)=2tanθ/(1-tanθ)

tan^2 θ=(1-cos⁡(2θ))/(1+cos⁡(2θ))

Half Angle

For sine

sin ∝/2=±√((1-cos∝)/2)

For cosine

cos ∝/2=±√((1+cos∝)/2)

For tangent

tan ∝/2=±√((1-cos∝)/(1+cos∝))

tan ∝/2=(1-cos∝)/(sin∝)

Sum to Product

For sine

sinα sinβ=1/2[cos⁡(α-β)-cos⁡(α+β)]

For cosine

cosα cosβ=1/2[cos⁡(α-β)+cos⁡(α+β)]

For sine and cosine

sinα cosβ=1/2[sin⁡(α+β)-sin⁡(α-β)]

Product to Sum

For sine

sinα+sinβ=2(sin (α+β)/2) (cos (α-β)/2)

sinα-sinβ=2(sin (α-β)/2) (cos (α+β)/2)

For cosine

cosα+cosβ=2(cos (α+β)/2) (cos (α-β)/2)

cosα-cosβ=-2(sin (α+β)/2) (sin (α-β)/2)

Equations

Radian Measure

θ=(arc length)/radius

θ=s/r

θ of an angle is the measure of the ratio of length of
the arc it spans on the circle to the length of the radius.

This can only be done in Radians

Arc Length Theorem

Formula in Radians

s=θr

Formula in Degrees

s=(θ/360)2πr

For a circle of radius r, a central angle (a positive angle whose
vertex is at the center of a circle) of θ radians subtends an arc whose length is s

Revolution of a unit circle

Formula in Radians

C=2π

Formula in Degrees

C=360

This is used to find the outside measure of a unit circle

Area of a Sector of a circle

A=1/2 r^2 θ

This can only be done in radians

The area of a sector of a circle is
proportional to the measure of the central angle.

Periodic Point

P=(x, y)

P=(cosθ, sinθ)

This is used to find points through the unit circle

Radius

x^2+y^2=r^2

For an angle in standard position, let P=(x,y) be the point on the
terminal side of the angle that is also on the circle .

Subtopic

Period of Trig Functions

Sine/Cosine/Cosecant/Secant

θ+2πk=θ

Their Period is 2π

Tangent/Cotangent

θ+πk=θ

Their Period is π

Even Properties

f(-θ)=θ

Odd Properties

f(-θ)=-f(θ)

Transformations

Normal

g(x)=af(b(x-h))+k

a is the vertical stretch/compression
b is the Horizontal stretch/compression
h is the horizontal shift and k is the vertical shift

Trig

f ( x )= A sin ( ωx−φ ) + B= A sin (ω (x− φ/ω ) )+ B

A is amplitude, ω is omega, φ is phi

Period: T=2π/ω

Horizontal Shift: ϕ/ω

Trigonometric Functions

Functions of an Angle

Circular Functions

Used to relate the angles of a triangle
to the lengths of the sides of a triangle

Important in Modeling of periodic Phenomena.

Uses Greek letter to denote angles

Alpha

Beta

Gamma

Theta

Primary Functions

sine function

sin

Asoociates each angle with the vertical
coordinate (y-coordinate)

Value within Points

In a Unit Circle

sinθ= y/1

Since in a unit circle the radius, or hypotenuse, is one, the result is y

In a Different Circles

sinαθ=y/r

Since the radius is more than one, y should
be divided by r

Measurements

1/2

π/6

√2/2

π/4

√3/2

π/3

π/2

180

-1

3π/2

270

2π

360

Properties

Input

Output

Domain

All Real Numbers

Range

[-1,1]

Inverse

sin^(-1) θ

x=siny
y=sin^(-1)x
arcsin

Domain

[-1,1]

Range

[-π/2,π/2]

cosine function

cos

Associates each angle with the horizontal
coordinate (x-coordinate)

Value within Points

In a Unit Circle

cosθ=x/1

Since in a unit circle the radius, or hypotenuse, is one, the result is x

In Different Circles

cosθ=x/r

Since the radius is more than one, x should
be divided by r

Measurements

√3/2

π/6

√2/2

π/4

1/2

π/3

π/2

-1

180

3π/2

270

2π

360

Properties

Input

Output

Domain

All Real Numbers

Range

[-1,1]

Inverse

cos^(-1) θ

x=cosy
y=cos^(-1)x
arccos

Domain

[-1,1]

Range

[0,π]

Tangent function

tanθ

sinθ/cosθ

Value within points

tanθ=x/y

Since radius is not neded to find tangent,
the equation is the same in all kind of circles

x≠0

Associates with the ratio of the y-coordinate
to the x-coordinate)

Measurement

√3/3

π/6

pi/4

√3

π/3

Undefined

π/2

180

Undefined

3π/2

270

2π

360

Properties

Input

any θ that does not produce division by zero

Output

y/x

Domain

All real numbers except odd integeres multiples of π/2

Range

All real numbers

Inverse

tan^(-1) θ

x=tany
y=tan^(-1)x
arctan

Domain

All real numbers

Range

-π/2<y<π/2

Reciprocal Functions

Cosecant Funtion

cscθ

1/sinθ

Value Within Points

In a unit circle

cscθ=1/y

Since in a unit circle the radius is one, the result is one over y

In Different Circles

cscθ=r/y

Since radius is more than one, radius should be divided by y

y≠0

Measurements

Undefined

π/6

√2

π/4

2√3/3

π/3

π/2

Undefined

180

-1

3π/2

270

Undefined

2π

360

Properties

Input

Any θ that does not produce division by zero

Output

1/y

Domain

All real numbers except integer multiples of π

Range

all real numbers greater than or equal to 1 or less than or equal to -1

Inverse

csc^(-1) θ

x=cscy
y=csc^(-1)x
arccsc

Domain

|x|≥1

Range

-π/2≤y≤π/2, y≠0

Secant Function

sec

1/cos

Value within Points

In a unit circle

cscθ=1/x

Since in a unit circle radius is one, the result is one over x

In Different Circles

cscθ=r/x

Since radius is more than one, the result is r divided by x

x≠0

Measurements

2√3/3

π/6

√2

π/4

π/3

Undefined

π/2

-1

180

Undefined

3π/2

270

2π

360

Properties

input

any θ

Output

1/x

Domain

All real numbers except odd integeres multiples of π/2

Range

all real numbers greater than or equal to 1 or less than or equal to -1

Inverse

sec^(-1) θ

x=secy
y=sec^(-1)x
arcsec

Domain

|x|≥1

Range

0≤y≤π, y≠ π/2

Cotangent Function

cot

1/tan

cos/sin

Value within Points

cotθ=x/y

Since radius is not needed to find cotangent,
the equation is the same in all kinds of circles

y≠0

Measurements

Undefined

√3

π/6

π/4

√3/3

π/3

π/2

Undefined

180

3π/2

270

Undefined

2π

360

Properties

Input

any θ

Output

x/y

Domain

All real numbers except integer multiples of π

Range

All real numbers

Inverse

cot^(-1) θ

x=coty
y=cot^(-1)x
arccot

Domain

All real numbers

Range

0<y<π, y≠ π/2

Trigonometric Equations

Solving equations is a technique that has been used since early Algebra courses.

It is used to find all x solutions

Solutions to equations are values of the variable
that make the equation a true statement.

Steps to solve

Identify the equation

See what function is being used

Observe the restrictions

cosθ=1, when ,0≤ θ ≤2π

Find the angle

cosθ=1

θ=0

Use the General Formula

θ+2kπ

0+2kπ

Helps you find the solutions as requested

One should know their angles and how to identify
when to stop due to restrictions.

k is any integer

Write solutions

x=(0,2π)

Always write as ordered pairs when is more than one solution)