Trigonometry

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Cell nucleus

Ved Felipe Miranda

Fundamentals of Geometry for Elementary School Teachers

Ved Hannah Schweiter

Quadratic Equations

Ved stefn dreaves

Quadratic Mind Map

Ved Patel Sheil

Trigonometry

Trigonometric Equations

Steps to solve

Write solutions

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

x=(0,2π)

Use the General Formula

k is any integer

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

Helps you find the solutions as requested

θ+2kπ

0+2kπ

Find the angle

cosθ=1

θ=0

Identify the equation

cosθ=1, when ,0≤ θ ≤2π

Observe the restrictions

See what function is being used

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

It is used to find all x solutions

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

Trigonometric Functions

Reciprocal Functions

Cotangent Function

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

cot^(-1) θ

Properties

x/y

any θ

cotθ=x/y

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

cot

cos/sin

1/tan

Secant Function

0≤y≤π, y≠ π/2

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

sec^(-1) θ

1/x

input

any θ

cscθ=r/x

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

cscθ=1/x

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

sec

1/cos

Cosecant Funtion

-π/2≤y≤π/2, y≠0

|x|≥1

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

csc^(-1) θ

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

All real numbers except integer multiples of π

1/y

Any θ that does not produce division by zero

2√3/3

√2

2

Value Within Points

y≠0

cscθ=r/y

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

In a unit circle

cscθ=1/y

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

cscθ

1/sinθ

Primary Functions

Tangent function

-π/2

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

tan^(-1) θ

All real numbers

All real numbers except odd integeres multiples of π/2

y/x

any θ that does not produce division by zero

Measurement

Undefined

√3

pi/4

√3/3

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

Value within points

x≠0

tanθ=x/y

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

tanθ

sinθ/cosθ

cosine function

[0,π]

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

cos^(-1) θ

x

In Different Circles

cosθ=x/r

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

cosθ=x/1

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

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

cos

sine function

Inverse

[-π/2,π/2]

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

sin^(-1) θ

Properties

Range

[-1,1]

Domain

All Real Numbers

Output

y

Input

Measurements

360

2π

-1

270

3π/2

180

π

1

90

π/2

√3/2

60

π/3

√2/2

45

π/4

1/2

30

π/6

0

Value within Points

In a Different Circles

sinαθ=y/r

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

In a Unit Circle

sinθ= y/1

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

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

sin

Uses Greek letter to denote angles

Theta

θ

Gamma

γ

Beta

β

Alpha

α

Important in Modeling of periodic Phenomena.

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

Circular Functions

Functions of an Angle

Equations

Transformations

Trig

Horizontal Shift: ϕ/ω

Period: T=2π/ω

A is amplitude, ω is omega, φ is phi

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

Normal

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

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

Odd Properties

f(-θ)=-f(θ)

Even Properties

f(-θ)=θ

Period of Trig Functions

Tangent/Cotangent

Their Period is π

θ+πk=θ

Sine/Cosine/Cosecant/Secant

Their Period is 2π

θ+2πk=θ

Radius

Subtopic

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 .

x^2+y^2=r^2

Periodic Point

This is used to find points through the unit circle

P=(cosθ, sinθ)

P=(x, y)

Area of a Sector of a circle

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

This can only be done in radians

A=1/2 r^2 θ

Revolution of a unit circle

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

C=360

C=2π

Arc Length Theorem

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

Formula in Degrees

s=(θ/360)2πr

Formula in Radians

s=θr

Radian Measure

This can only be done in Radians

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.

θ=s/r

θ=(arc length)/radius

Trigonometric Identities

Product to Sum

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

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

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

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

Sum to Product

For sine and cosine

sinα cosβ=1/2[sin⁡(α+β)-sin⁡(α-β)]

cosα cosβ=1/2[cos⁡(α-β)+cos⁡(α+β)]

For sine

sinα sinβ=1/2[cos⁡(α-β)-cos⁡(α+β)]

Half Angle

tan ∝/2=(1-cos∝)/(sin∝)

tan ∝/2=±√((1-cos∝)/(1+cos∝))

cos ∝/2=±√((1+cos∝)/2)

sin ∝/2=±√((1-cos∝)/2)

Double Angle

tan^2 θ=(1-cos⁡(2θ))/(1+cos⁡(2θ))

tan⁡(2θ)=2tanθ/(1-tanθ)

cos^2 θ=(1+cos⁡(2θ))/2

cos⁡(2θ)=2cos^2 θ-1

cos⁡(2θ)=1-2sin^2 θ

cos⁡(2θ)=cos^2 θ+sin^2 θ

sin^2 θ=1-cos⁡(2θ)/2

sin⁡(2θ)=2sinθcosθ

Sum/Difference

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) )

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 cosine

cos⁡(a-b)=cos⁡a cos⁡b+sin⁡a sin⁡b

cos⁡(a+b)=cos⁡a cos⁡b-sin⁡a sin⁡b

Even

Evens with a negative angle results as positive

sec(-θ)=secθ

cos(-θ)=cosθ

Odd

Odds with a negative angle results as negative

cot(-θ)=-cotθ

tan(-θ)=-tanθ

csc(-θ)=-cscθ

sin⁡(-θ)=-sinθ

Pythagorean

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

cscθ^2-cotθ^2=1

secθ^2-tanθ^2=1

sinθ^2+cosθ^2=1

Reciprocal

Each primary function has a reciprocal identity

cotθ=1/tanθ

secθ=1/cosθ

cscθ=1/sinθ

Quotient

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

cotθ=cosθ/sinθ

tanθ=sinθ/cosθ