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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.
$P(t)=950-450\mathrm{sin}\left(\frac{\pi}{6}t\right)$
Daily temperatures in the desert can be very extreme. If the temperature varies from $\text{\hspace{0.17em}}90\text{\xb0F}\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}30\text{\xb0F}\text{\hspace{0.17em}}$ 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.
$y=3\text{\hspace{0.17em}}\mathrm{cos}(x\pi )$
Amplitude: 3, period: 2, frequency: $\frac{1}{2}$ Hz
$y=\mathrm{-2}\text{\hspace{0.17em}}\mathrm{sin}(16x\pi )$
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.
$C(t)=20\mathrm{sin}(2\pi t)+100{(1.4427)}^{t}$
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.
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
$\mathrm{sin}(-x)\mathrm{cos}(-2x)-\mathrm{sin}(-x)\mathrm{cos}(-2x)$
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 $\text{\hspace{0.17em}}[0,2\pi ).$
${\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1=0$
$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 $\text{\hspace{0.17em}}\mathrm{tan}(x)-\sqrt{3}=0.$
$\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.
$\frac{1}{8}\left(3+\mathrm{cos}(4x)-4\mathrm{cos}(2x)\right)$
For the following exercises, prove the identity.
${\mathrm{tan}}^{3}x-\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}{\mathrm{sec}}^{2}x=\mathrm{tan}\left(-x\right)$
$\mathrm{sin}\left(3x\right)-\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)={\mathrm{cos}}^{2}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x-{\mathrm{sin}}^{3}x$
$$\begin{array}{r}\hfill \mathrm{sin}(3x)-\mathrm{cos}x\mathrm{sin}(2x)=\text{}\\ \hfill \mathrm{sin}(x+2x)-\mathrm{cos}x(2\mathrm{sin}x\mathrm{cos}x)=\text{}\\ \hfill \mathrm{sin}x\mathrm{cos}(2x)+\mathrm{sin}(2x)\mathrm{cos}x-2\mathrm{sin}x{\mathrm{cos}}^{2}x=\text{}\\ \hfill \mathrm{sin}x({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x)+2\mathrm{sin}x\mathrm{cos}x\mathrm{cos}x-2\mathrm{sin}x{\mathrm{cos}}^{2}x=\text{}\\ \hfill \mathrm{sin}x{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{3}+0=\text{}\\ \hfill {\mathrm{cos}}^{2}x\mathrm{sin}x-{\mathrm{sin}}^{3}x={\mathrm{cos}}^{2}x\mathrm{sin}x-{\mathrm{sin}}^{3}x\end{array}$$
$\frac{\mathrm{sin}\left(2x\right)}{\mathrm{sin}\text{\hspace{0.17em}}x}-\frac{\mathrm{cos}\left(2x\right)}{\mathrm{cos}\text{\hspace{0.17em}}x}=\mathrm{sec}\text{\hspace{0.17em}}x$
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.
$x$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ |
$y$ | $\mathrm{-2}$ | $2$ | $\mathrm{-2}$ | $2$ | $\mathrm{-2}$ | $2$ |
$y=2\mathrm{cos}\left(\pi x+\pi \right)$
The displacement $\text{\hspace{0.17em}}h(t)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h(t)=\frac{1}{4}\text{\hspace{0.17em}}\mathrm{sin}(120\pi t),$ 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 $\text{\hspace{0.17em}}n(t)=8\text{\hspace{0.17em}}\mathrm{cos}(20\pi t)\mathrm{cos}(1000\pi t).\text{\hspace{0.17em}}$ 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?
$6+5\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi}{6}(1-x)\right)\text{\hspace{0.17em}}$ . 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?
$D(t)=2\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi}{6}t\right)+108+\frac{1}{4}t,$ 93.5855 months (or 7.8 years) from now
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