# 3.6 Modeling with trigonometric equations  (Page 12/14)

 Page 12 / 14

For the following exercises, construct functions that model the described behavior.

A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to $\text{\hspace{0.17em}}t,$ the month.

$P\left(t\right)=950-450\mathrm{sin}\left(\frac{\pi }{6}t\right)$

Daily temperatures in the desert can be very extreme. If the temperature varies from $\text{\hspace{0.17em}}90\text{°F}\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}30\text{°F}\text{\hspace{0.17em}}$ and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.

For the following exercises, find the amplitude, frequency, and period of the given equations.

$y=3\text{\hspace{0.17em}}\mathrm{cos}\left(x\pi \right)$

Amplitude: 3, period: 2, frequency: $\frac{1}{2}$ Hz

$y=-2\text{\hspace{0.17em}}\mathrm{sin}\left(16x\pi \right)$

For the following exercises, model the described behavior and find requested values.

An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to $\text{\hspace{0.17em}}t,$ the number of years from now.

$C\left(t\right)=20\mathrm{sin}\left(2\pi t\right)+100{\left(1.4427\right)}^{t}$

The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to $\text{\hspace{0.17em}}t,$ the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.

## Practice test

For the following exercises, simplify the given expression.

$\mathrm{cos}\left(-x\right)\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}x+{\mathrm{sin}}^{2}x$

1

$\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-2x\right)-\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-2x\right)$

For the following exercises, find the exact value.

$\mathrm{cos}\left(\frac{7\pi }{12}\right)$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

$\mathrm{tan}\left(\frac{3\pi }{8}\right)$

$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{tan}}^{-1}\sqrt{3}\right)$

$-\sqrt{2}-\sqrt{3}$

$2\mathrm{sin}\left(\frac{\pi }{4}\right)\mathrm{sin}\left(\frac{\pi }{6}\right)$

For the following exercises, find all exact solutions to the equation on $\text{\hspace{0.17em}}\left[0,2\pi \right).$

${\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1=0$

$0,\pi$

${\mathrm{cos}}^{2}x=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-3=0$

${\mathrm{sin}}^{-1}\left(\frac{1}{4}\left(\sqrt{13}-1\right)\right),\pi -{\mathrm{sin}}^{-1}\left(\frac{1}{4}\left(\sqrt{13}-1\right)\right)$

$\mathrm{cos}\left(2x\right)+{\mathrm{sin}}^{2}x=0$

$2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-\mathrm{sin}\text{\hspace{0.17em}}x=0$

$0,\frac{\pi }{6},\frac{5\pi }{6},\pi$

Rewrite the expression as a product instead of a sum: $\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)+\mathrm{cos}\left(-8x\right).$

Find all solutions of $\text{\hspace{0.17em}}\mathrm{tan}\left(x\right)-\sqrt{3}=0.$

$\frac{\pi }{3}+k\pi$

Find the solutions of $\text{\hspace{0.17em}}{\mathrm{sec}}^{2}x-2\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=15\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right)\text{\hspace{0.17em}}$ algebraically; then graph both sides of the equation to determine the answer.

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(2\theta \right)\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\theta =-\frac{3}{4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is on the interval $\text{\hspace{0.17em}}\left[\frac{\pi }{2},\pi \right].$

$\text{\hspace{0.17em}}-\frac{24}{25},-\frac{7}{25},\frac{24}{7}$

Find $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\theta }{2}\right),\mathrm{cos}\left(\frac{\theta }{2}\right),$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\theta }{2}\right)\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{7}{25}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in quadrant IV.

Rewrite the expression $\text{\hspace{0.17em}}{\mathrm{sin}}^{4}x\text{\hspace{0.17em}}$ with no powers greater than 1.

$\frac{1}{8}\left(3+\mathrm{cos}\left(4x\right)-4\mathrm{cos}\left(2x\right)\right)$

For the following exercises, prove the identity.

${\mathrm{tan}}^{3}x-\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}{\mathrm{sec}}^{2}x=\mathrm{tan}\left(-x\right)$

$\mathrm{sin}\left(3x\right)-\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)={\mathrm{cos}}^{2}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-{\mathrm{sin}}^{3}x$

$\frac{\mathrm{sin}\left(2x\right)}{\mathrm{sin}\text{\hspace{0.17em}}x}-\frac{\mathrm{cos}\left(2x\right)}{\mathrm{cos}\text{\hspace{0.17em}}x}=\mathrm{sec}\text{\hspace{0.17em}}x$

Plot the points and find a function of the form $\text{\hspace{0.17em}}y=A\mathrm{cos}\left(Bx+C\right)+D\text{\hspace{0.17em}}$ that fits the given data.

 $x$ $0$ $1$ $2$ $3$ $4$ $5$ $y$ $-2$ $2$ $-2$ $2$ $-2$ $2$

$y=2\mathrm{cos}\left(\pi x+\pi \right)$

The displacement $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h\left(t\right)=\frac{1}{4}\text{\hspace{0.17em}}\mathrm{sin}\left(120\pi t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15 th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.

${81.5}^{\circ },{78.7}^{\circ }$

Two frequencies of sound are played on an instrument governed by the equation $\text{\hspace{0.17em}}n\left(t\right)=8\text{\hspace{0.17em}}\mathrm{cos}\left(20\pi t\right)\mathrm{cos}\left(1000\pi t\right).\text{\hspace{0.17em}}$ What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?

The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?

$6+5\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}\left(1-x\right)\right)\text{\hspace{0.17em}}$ . From November 23 to February 6.

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.

Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?

$D\left(t\right)=2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}t\right)+108+\frac{1}{4}t,$ 93.5855 months (or 7.8 years) from now

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers! By OpenStax By Courntey Hub By Edgar Delgado By OpenStax By Stephen Voron By OpenStax By Jonathan Long By Robert Murphy By Tod McGrath By Brooke Delaney