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What are the amplitude, period, and phase shift for the function?

amplitude: 8,000; period: 10; phase shift: 0

Over this domain, when does the population reach 18,000? 13,000?

What is the predicted population in 2007? 2010?

In 2007, the predicted population is 4,413. In 2010, the population will be 11,924.

For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

Suppose the graph of the displacement function is shown in [link] , where the values on the x -axis represent the time in seconds and the y -axis represents the displacement in inches. Give the equation that models the vertical displacement of the weight on the spring.

A graph of a consine function over one period. Graphed on the domain of [0,10]. Range is [-5,5].

At time = 0, what is the displacement of the weight?

5 in.

At what time does the displacement from the equilibrium point equal zero?

What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?

10 seconds

Inverse Trigonometric Functions

For the following exercises, find the exact value without the aid of a calculator.

sin 1 ( 1 )

cos 1 ( 3 2 )

π 6

tan −1 ( −1 )

cos 1 ( 1 2 )

π 4

sin 1 ( 3 2 )

sin 1 ( cos ( π 6 ) )

π 3

cos 1 ( tan ( 3 π 4 ) )

sin ( sec 1 ( 3 5 ) )

No solution

cot ( sin 1 ( 3 5 ) )

tan ( cos 1 ( 5 13 ) )

12 5

sin ( cos 1 ( x x + 1 ) )

Graph f ( x ) = cos x and f ( x ) = sec x on the interval [ 0 , 2 π ) and explain any observations.

The graphs are not symmetrical with respect to the line y = x . They are symmetrical with respect to the y -axis.

A graph of cosine of x and secant of x. Cosine of x has maximums where secant has minimums and vice versa. Asymptotes at x=-3pi/2, -pi/2, pi/2, and 3pi/2.

Graph f ( x ) = sin x and f ( x ) = csc x and explain any observations.

Graph the function f ( x ) = x 1 x 3 3 ! + x 5 5 ! x 7 7 ! on the interval [ 1 , 1 ] and compare the graph to the graph of f ( x ) = sin x on the same interval. Describe any observations.

The graphs appear to be identical.

Two graphs of two identical functions on the interval [-1 to 1]. Both graphs appear sinusoidal.

Chapter practice test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

f ( x ) = 0.5 sin x

amplitude: 0.5; period: 2 π ; midline y = 0

A graph of two periods of a sinusoidal function, graphed over -2pi to 2pi. The range is [-0.5,0.5]. X-intercepts at multiples of pi.

f ( x ) = 5 cos x

f ( x ) = 5 sin x

amplitude: 5; period: 2 π ; midline: y = 0

Two periods of a sine function, graphed over -2pi to 2pi. The range is [-5,5], amplitude of 5, period of 2pi.

f ( x ) = sin ( 3 x )

f ( x ) = cos ( x + π 3 ) + 1

amplitude: 1; period: 2 π ; midline: y = 1

A graph of two periods of a cosine function, graphed over -7pi/3 to 5pi/3. Range is [0,2], Period is 2pi, amplitude is1.

f ( x ) = 5 sin ( 3 ( x π 6 ) ) + 4

f ( x ) = 3 cos ( 1 3 x 5 π 6 )

amplitude: 3; period: 6 π ; midline: y = 0

A graph of two periods of a cosine function, over -7pi/2 to 17pi/2. The range is [-3,3], period is 6pi, and amplitude is 3.

f ( x ) = tan ( 4 x )

f ( x ) = 2 tan ( x 7 π 6 ) + 2

amplitude: none; period:   π ;   midline:   y = 0 , asymptotes:   x = 2 π 3 + π k , where   k   is an integer

A graph of two periods of a tangent function over -5pi/6 to 7pi/6. Period is pi, midline at y=0.

f ( x ) = π cos ( 3 x + π )

f ( x ) = 5 csc ( 3 x )

amplitude: none; period:   2 π 3 ;   midline:   y = 0 , asymptotes:   x = π 3 k , where   k   is an integer

A graph of two periods of a cosecant functinon, over -2pi/3 to 2pi/3. Vertical asymptotes at multiples of pi/3. Period of 2pi/3.

f ( x ) = π sec ( π 2 x )

f ( x ) = 2 csc ( x + π 4 ) 3

amplitude: none; period: 2 π ; midline: y = 3

A graph of two periods of a cosecant function, graphed from -9pi/4 to 7pi/4. Period is 2pi, midline at y=-3.

For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed from -2 to 2. Range is [-6,-2], period is 2, and amplitude is 2.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed over -2 to 2. Range is [-2,2], period is 2, and amplitude is 2.

amplitude: 2; period: 2; midline: y = 0 ; f ( x ) = 2 sin ( π ( x 1 ) )

Give in terms of a tangent function.

A graph of two periods of a tangent function, graphed over -3pi/4 to 5pi/4. Vertical asymptotes at x=-pi/4, 3pi/4. Period is pi.

For the following exercises, find the amplitude, period, phase shift, and midline.

y = sin ( π 6 x + π ) 3

amplitude: 1; period: 12; phase shift: −6 ; midline y = −3

y = 8 sin ( 7 π 6 x + 7 π 2 ) + 6

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D , in terms of t .

D ( t ) = 68 12 sin ( π 12 x )

Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

For the following exercises, find the period and horizontal shift of each function.

g ( x ) = 3 tan ( 6 x + 42 )

period: π 6 ; horizontal shift: −7

n ( x ) = 4 csc ( 5 π 3 x 20 π 3 )

Write the equation for the graph in [link] in terms of the secant function and give the period and phase shift.

A graph of 2 periods of a secant function, graphed over -2 to 2. The period is 2 and there is no phase shift.

f ( x ) = sec ( π x ) ; period: 2; phase shift: 0

If tan x = 3 , find tan ( x ) .

If sec x = 4 , find sec ( x ) .

4

For the following exercises, graph the functions on the specified window and answer the questions.

Graph m ( x ) = sin ( 2 x ) + cos ( 3 x ) on the viewing window [ 10 , 10 ] by [ 3 , 3 ] . Approximate the graph’s period.

Graph n ( x ) = 0.02 sin ( 50 π x ) on the following domains in x : [ 0 , 1 ] and [ 0 , 3 ] . Suppose this function models sound waves. Why would these views look so different?

The views are different because the period of the wave is 1 25 . Over a bigger domain, there will be more cycles of the graph.

Two side-by-side graphs of a sinusodial function. The first graph is graphed over 0 to 1, the second graph is graphed over 0 to 3. There are many periods for each.

Graph f ( x ) = sin x x on [ 0.5 , 0.5 ] and explain any observations.

For the following exercises, let f ( x ) = 3 5 cos ( 6 x ) .

What is the largest possible value for f ( x ) ?

3 5

What is the smallest possible value for f ( x ) ?

Where is the function increasing on the interval [ 0 , 2 π ] ?

On the approximate intervals ( 0.5 , 1 ) , ( 1.6 , 2.1 ) , ( 2.6 , 3.1 ) , ( 3.7 , 4.2 ) , ( 4.7 , 5.2 ) , ( 5.6 , 6.28 )

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

Sine curve with amplitude 3, period π 3 , and phase shift ( h , k ) = ( π 4 , 2 )

Cosine curve with amplitude 2, period π 6 , and phase shift ( h , k ) = ( π 4 , 3 )

f ( x ) = 2 cos ( 12 ( x + π 4 ) ) + 3

A graph of one period of a cosine function, graphed over -pi/4 to 0. Range is [1,5], period is pi/6.

For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

f ( x ) = 5 cos ( 3 x ) + 4 sin ( 2 x )

f ( x ) = e sin t

This graph is periodic with a period of 2 π .

A graph of two periods of a sinusoidal function, The graph has a period of 2pi.

For the following exercises, find the exact value.

sin 1 ( 3 2 )

tan 1 ( 3 )

π 3

cos 1 ( 3 2 )

cos 1 ( sin ( π ) )

π 2

cos 1 ( tan ( 7 π 4 ) )

cos ( sin 1 ( 1 2 x ) )

1 ( 1 2 x ) 2

cos 1 ( 0.4 )

cos ( tan 1 ( x 2 ) )

1 1 + x 4

For the following exercises, suppose sin t = x x + 1 .

tan t

csc t

x + 1 x

Given [link] , find the measure of angle θ to three decimal places. Answer in radians.

An illustration of a right triangle with angle theta. Opposite the angle theta is a side with length 12, adjacent to the angle theta is a side with length 19.

For the following exercises, determine whether the equation is true or false.

arcsin ( sin ( 5 π 6 ) ) = 5 π 6

False

arccos ( cos ( 5 π 6 ) ) = 5 π 6

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.

approximately 0.07 radians

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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