Trigonometry

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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