Trig Functions and Identities Name - Amazon S3€¦ · Trig Functions and Identities Name: Notes Date: Nicole is studying the effects of algae blooms on a coral reef. Algae typically

Trig Functions and Identities Name:

Notes Date:

Nicole is studying the effects of algae blooms on a coral reef. Algae

typically can be modeled using a sinusoidal function. Using the

amplitude, period and frequency Nicole needs to choose an algae she

can observe through a number of bloom cycles. If the bloom cycle is

too short or too large, it is challenging to observe in detail. If the

bloom cycle is too long, the study will have a number of other

variables introduced. She must decide which algae to observe.

Write the definition to the term and include an image or example that represents it.

Term Definition Example

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent

Term Definition Example

Unit Circle

Pythagorean

Identity

Amplitude

Period

Frequency

What are Trig Ratios? The Conceptualizer!

SOH CAH TOA is likely burned into your

math brain. It is a mnemonic device for

three of the most common trig relationships

for an angle in a right triangle.

Wait -- what? There are more?

There are six trig relationships in all:

ines = oppositehypotenuse osecantc = opposite

hypotenuse

osinec = adjacenthypotenuse ecants = adjacent

hypotenuse

angentt = oppositeadjacent otangentc = opposite

adjacent

© Clark Creative Education

Trig Ratios as Trig Functions The Conceptualizer!

We have put values into sine, cosine and

tangent and gotten values back -- you may

have done this before you realized they

were functions -- just like lines,

exponentials and quadratics.

Recall, with the unit circle that the sine

was the y value of a point on a circle,

cosine was the x value and tangent was the

y value divided by the x value.

Trig Ratios as Trig Functions The Conceptualizer!

When we start graphing trig functions, the

expectation is that the values entered will

be in RADIANS.

Recall --

Degrees are measured using 360 degree

measurement system (arbitrarily created by

the Babylonians) whereas radians are

measured relative to the radius of the

circle.

Radians are needed when performing

calculations because they are based on

what we know of mathematical concepts

like circumference.


Graphing Trigonometric Functions The Conceptualizer!

From the right triangle diagrams on the unit

circle for sine we can see --

-- as the angle (in Quadrant I) gets larger,

the opposite side gets larger until it reaches

1 --

-- then it moves into Quadrant II where it

gets smaller until it reaches 0 --

-- then when it moves into Quadrant III

where it gets negative.

#moreFunwithGIFs

Amplitude, Period & Frequency The Conceptualizer!

Oscillating, sine and cosine functions have

characteristics that are used to better

understand them.

Amplitude is the height between the center

line and the top of the function (or the

bottom). Alternatively, the amplitude is

half the difference between the maximum

and minimum, and half of 2 is 1.

Then the period of the sine curve is how

long it takes to complete a cycle.

The frequency is the number of complete

cycles in an interval It is the reciprocal of

the period.

requencyf = 1period


Amplitude, Period & Frequency Notes

Identify the amplitude, period and frequency from this graph:

Transforming Trig Functions The Conceptualizer!

All trig functions can be written in the form

(replacing sine as appropriate):

in(bx ) y = a · s + c + d

Each of these variables transforms different

aspects of the functions and affects the

graph’s amplitude, period and frequency.

Notice a and d are “outside” of the function

and b and c are “inside” of the function.

Outside elements affect the function

vertically and inside elements affect the

function horizontally.

We will begin with a, which affects the

graph’s amplitude.

A function of the form has an in(x)y = a · s

amplitude of ; and if a is negative, the a||

graph is reflected across the x axis.



What happens if you add a constant to the y

value, after other calculations have been

done? This affects the d in the general

equation.

A function of the form also in(x) y = a · s + d

shows a vertical shift by d units.

Importantly, this shifts the center line,

which may impact how you identify the

amplitude graphically.

Transforming Trig Functions Notes

Describe the graph of − sin(x) y = 2 − 3compared to .in(x)y = s


What happens if you multiply the x value,

before other calculations have been done?

This affects the b in the general equation.

Then a function of the form:

in(bx) y = a · s + d

shows a horizontal compression or

expansion with a factor of b. The period is

.And the frequency is .b2π b

2π


Language Police!

A “b” value that is larger than 1 makes the graph appear squeezed or compressed

horizontally, with a higher frequency and a shorter period. A “b” value between -1 and 1,

such as 1/2 makes the graph appear stretched out horizontally, wider, with a longer period

and a lower frequency.


There is one last transformation for

sinusoidal functions: a horizontal shift that

impacts the c variable.

We use the term phase shift when talking

about horizontal shifts in sinusoidal

functions.

A positive c shows a left shift, while a

negative c shows a right shift.

Then a function of the form

in(bx ) y = a · s + c + d

also shows a horizontal shift, by .cb


Transformation Station

These function transformations we’ve discussed apply to all functions.

(with a little tweak on the c variable)

That is, the graph of shows the same transformations from − sin(3x)y = 5 + 4

the parent function that shows, compared to its in(x)y = s − |3x|y = 5 + 4

parent function .x|y = |


Describe the graph of

− cos(5x )y = 41 + 2

π + 4

and sketch it.



Write the cosine function with an amplitude

of 2, a period of , a horizontal right shift 3π

of , and a vertical shift of 4. Sketch the π12

function.

Reciprocal Identities Extra! Extra!

Trig functions aren’t just for graphing, but

they also are for simplifying complex

expressions.

Recall from earlier, the secant is the

hypotenuse, over the adjacent side.

ecants = adjacenthypotenuse

But this is just the reciprocal of cosine.


Using Reciprocal Identities Notes

If , what is ?ec(x)s = 45 an(x)t

Quotient Identities Extra! Extra!

We have known for a long an(x)t = sin(x)cos(x)

time. Now we know we should call it an

identity. These are called “quotient

identities” or “ratio identities”.

Pythagorean Identities The Conceptualizer!

The Pythagorean Theorem relates the three

sides of a right triangle. This is leads to

“Pythagorean Identities”:

x xsin2 + cos2 = 1

You’ll also see it in disguise as:

x sin xcos2 = 1 − 2

x cos xsin2 = 1 − 2

If you divide the entire identity by xsin2

and use reciprocal identities you will get:

x x1 + cot2 = csc2

If you divide the entire identity by xcos2

and use reciprocal identities you will get:

x xtan2 + 1 = sec2

These are Pythagorean Identities too.


Proving an Identity Notes

Show that x x1 + tan2 = sec2

Using Identities Notes

Show that in(x)cot(x) os(x)s = c

Using Identities Notes

Show that equals1 an(x))(1 ot(x))( + t − c

an(x) ot(x)t − c


List the identities into the appropriate categories.

Reciprocal Identities Quotient Identities Pythagorean Identities

There are a few other useful identities to be aware of to use as reference.

Co-Function Identities (Phase Change) Sum-Difference Identities

Shifting trig functions can result in other functions.

in( ) cos(u)s 2π − u = os( ) sin(u)c 2

π − u =

an( ) cot(u)t 2π − u = sc( ) sec(u)c 2

π − u =

ec( ) csc(u)s 2π − u = ot( ) tan(u)c 2

π − u =

in(u ) sin(u)cos(v) os(u)sin(v)s ± v = ± c

os(u ) cos(u)cos(v) in(u)sin(v)c ± v = ∓ s

an(u )t ± v = tan(u)±tan(v)1∓tan(u)tan(v)

Double Angle Identities Half Angle Identities

in(2u) sin(u)cos(u)s = 2

os(2u) (u) (u)c = cos2 − sin2

an(2u)t = 2tan(u)1−tan (u)2

(u)sin2 = 21−cos(2u)

(u)cos2 = 21+cos(2u)

(u)tan2 = 1+cos(2u)1−cos(2u)


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Trig Functions and Identities Name - Amazon S3€¦ · Trig Functions and Identities Name: Notes Date: Nicole is studying the effects of algae blooms on a coral reef. Algae typically