7.6 Modeling with Trigonometric Functions

What are the amplitude and period of the function $y = 3 \cos \left(3 \pi x\right) ?$

The following equation represents a damped harmonic motion model: $f \left(t\right) = 5 e^{- 6 t} \cos \left(4 t\right)$ Find the initial displacement, the damping constant, and the frequency.

Write the equation for damped harmonic motion given $a = 10 , c = 0.5 ,$ and $p = 2.$

A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time $t = 0$ and allowed to oscillate. After $\frac{1}{3}$ second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.

A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.

Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?

Find a possible formula for the trigonometric function represented by the given table of values.

Find a possible formula for the trigonometric function represented by the given table of values.

Find a possible formula for the trigonometric function represented by the given table of values.

Find a possible formula for the trigonometric function represented by the given table of values.

Find a possible formula for the trigonometric function represented by the given table of values.

Construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $105 °\text{F}$ occurs at 5PM and the average temperature for the day is $85 °\text{F} .$ Find the temperature, to the nearest degree, at 9AM.

Construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $47 °\text{F}$ and $63 °\text{F}$ during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach $51 °\text{F}?$

Construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

A Ferris wheel is 20 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How much of the ride, in minutes and seconds, is…

Construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

The sea ice area around the North Pole fluctuates between about 6 million square kilometers on September 1 to 14 million square kilometers on March 1. Assuming a sinusoidal fluctuation, when are there less than 9 million square kilometers of sea ice? Give your answer as a range of da…

Construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. If the rainfall fluctuates between a low of 2 inches on day 10 and 12 inches on day 55, during what period is daily rainfall more than 10 inches?

Construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

In a certain region, monthly precipitation peaks at 8 inches on June 1 and falls to a low of 1 inch on December 1. Identify the periods when the region is under flood conditions (greater than 7 inches) and drought conditions (less than 2 inches). Give your answer in terms of the near…

Find the amplitude, period, and frequency of the given function.

The displacement $h \left(t\right)$ in centimeters of a mass suspended by a spring is modeled by the function $h \left(t\right) = 8 \sin \left(6 \pi t\right) ,$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Find the amplitude, period, and frequency of the given function.

The displacement $h \left(t\right)$ in centimeters of a mass suspended by a spring is modeled by the function $h \left(t\right) = 4 \cos \left(\frac{\pi}{2} t\right) ,$ where $t$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.

Construct an equation that models the described behavior.

A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, $P ,$ in terms of months since January, $t .$

Construct an equation that models the described behavior.

A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find a function that models the population, $P ,$ in terms of months since January, $t .$

Construct an equation that models the described behavior.

A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find a function that models the distance, $D ,$ the end of the spring is from equilibrium in terms of seconds, $t ,$ since the spring was released.

Construct an equation that models the described behavior.

A spring attached to the ceiling is pulled 17 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, $D ,$ the end of the spring is from equilibrium in terms of seconds, $t ,$ since the spring was released.

Create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A certain lake currently has an average trout population of 20,000. The population naturally oscillates above and below average by 2,000 every year. This year, the lake was opened to fishermen. If fishermen catch 3,000 fish every year, how long will it take for the lake to have no more trout?

Create a function modeling the described behavior. Then, calculate the desired result using a calculator.

A spring attached to a ceiling is pulled down 11 cm from equilibrium and released. After 2 seconds, the amplitude has decreased to 6 cm. The spring oscillates 8 times each second. Find when the spring first comes between $- 0.1$ and $0.1 \textrm{ }\text{cm},$ effectively at rest.

Create a function modeling the described behavior. Then, calculate the desired result using a calculator.

Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 8 times per second, was initially pulled down 32 cm from equilibrium, and the amplitude decreases by 50% each second. The second spring, oscillating 18 times per second, was initially…

A plane flies 1 hour at 150 mph at $22^{\circ}$ east of north, then continues to fly for 1.5 hours at 120 mph, this time at a bearing of $112^{\circ}$ east of north. Find the total distance from the starting point and the direct angle flown north of east.

Find a function of the form $y = a b^{x} + c \sin \left(\frac{\pi}{2} x\right)$ that fits the given data.

Find a function of the form $y = a b^{x} + c \sin \left(\frac{\pi}{2} x\right)$ that fits the given data.

Find a function of the form $y = a b^{x} \cos \left(\frac{\pi}{2} x\right) + c$ that fits the given data.

Find all solutions exactly that exist on the interval $\left[0 , 2 \pi\right) .$

$2 \sin \theta = - 1$

Find the exact value.

$\sin \left(70^{\circ}\right) \cos \left(25^{\circ}\right) - \cos \left(70^{\circ}\right) \sin \left(25^{\circ}\right)$

Find the exact value.

Find $\sin \left(2 \theta\right) , \cos \left(2 \theta\right) ,$ and $\tan \left(2 \theta\right)$ given $\sec \theta = - \frac{5}{3}$ and $\theta$ is in the interval $\left[\frac{\pi}{2} , \pi\right] .$

$\sin \left(\frac{\beta}{2}\right) , \cos \left(\frac{\beta}{2}\right) , \tan \left(\frac{\beta}{2}\right) , \sin \left(\frac{\alpha}{2}\right) , \cos \left(\frac{\alpha}{2}\right) , \textrm{ }\text{and}\textrm{ } \tan \left(\frac{\alpha}{2}\right)$

Graph each side of the equation to find the zeroes on the interval $\left[0 , 2 \pi\right) .$

$\sec^{2} x - 2 \sec x = 15$

Graph the points and find a possible formula for the trigonometric values in the given table.

Graph the points and find a possible formula for the trigonometric values in the given table.

A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

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 $t ,$ the month.

Find the amplitude, frequency, and period of the given equations.

$y = 3 \cos \left(x \pi\right)$

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 $t ,$ the number of years from now.

Find all exact solutions to the equation on $\left[0 , 2 \pi\right) .$

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

Find all exact solutions to the equation on $\left[0 , 2 \pi\right) .$

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