# 7.6 Modeling with trigonometric equations  (Page 11/14)

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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

what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil