# 7.6 Modeling with trigonometric equations  (Page 10/14)

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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 pulled down 15 cm from equilibrium and after 4 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than

Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than

Spring 2 comes to rest first after 8.0 seconds.

## Extensions

A plane flies 1 hour at 150 mph at $\text{\hspace{0.17em}}{22}^{\circ }\text{\hspace{0.17em}}$ east of north, then continues to fly for 1.5 hours at 120 mph, this time at a bearing of $\text{\hspace{0.17em}}{112}^{\circ }\text{\hspace{0.17em}}$ east of north. Find the total distance from the starting point and the direct angle flown north of east.

A plane flies 2 hours at 200 mph at a bearing of then continues to fly for 1.5 hours at the same speed, this time at a bearing of $\text{\hspace{0.17em}}{150}^{\circ }.\text{\hspace{0.17em}}$ Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.

500 miles, at $\text{\hspace{0.17em}}{90}^{\circ }$

For the following exercises, find a function of the form $\text{\hspace{0.17em}}y=a{b}^{x}+c\mathrm{sin}\left(\frac{\pi }{2}x\right)\text{\hspace{0.17em}}$ that fits the given data.

 $x$ 0 1 2 3 $y$ 6 29 96 379
 $x$ 0 1 2 3 $y$ 6 34 150 746

$y=6{\left(5\right)}^{x}+4\mathrm{sin}\left(\frac{\pi }{2}x\right)$

 $x$ 0 1 2 3 $y$ 4 0 16 -40

For the following exercises, find a function of the form $\text{\hspace{0.17em}}y=a{b}^{x}\mathrm{cos}\left(\frac{\pi }{2}x\right)+c\text{\hspace{0.17em}}$ that fits the given data.

 $x$ 0 1 2 3 $y$ 11 3 1 3

$y=8{\left(\frac{1}{2}\right)}^{x}\mathrm{cos}\left(\frac{\pi }{2}x\right)+3$

 $x$ 0 1 2 3 $y$ 4 1 −11 1

## Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right).$

${\mathrm{csc}}^{2}t=3$

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

${\mathrm{cos}}^{2}x=\frac{1}{4}$

$2\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =-1$

$\frac{7\pi }{6},\frac{11\pi }{6}$

$\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x+\mathrm{sin}\left(-x\right)=0$

$9\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\omega -2=4\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\omega$

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

$1-2\text{\hspace{0.17em}}\mathrm{tan}\left(\omega \right)={\mathrm{tan}}^{2}\left(\omega \right)$

For the following exercises, use basic identities to simplify the expression.

$\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x+\mathrm{cos}\text{\hspace{0.17em}}x-\frac{1}{\mathrm{sec}\text{\hspace{0.17em}}x}$

$1$

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

For the following exercises, determine if the given identities are equivalent.

${\mathrm{sin}}^{2}x+{\mathrm{sec}}^{2}x-1=\frac{\left(1-{\mathrm{cos}}^{2}x\right)\left(1+{\mathrm{cos}}^{2}x\right)}{{\mathrm{cos}}^{2}x}$

Yes

${\mathrm{tan}}^{3}x\text{\hspace{0.17em}}{\mathrm{csc}}^{2}x\text{\hspace{0.17em}}{\mathrm{cot}}^{2}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x=1$

## Sum and Difference Identities

For the following exercises, find the exact value.

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

$-2-\sqrt{3}$

$\mathrm{cos}\left(\frac{25\pi }{12}\right)$

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

$\frac{\sqrt{2}}{2}$

$\mathrm{cos}\left({83}^{\circ }\right)\mathrm{cos}\left({23}^{\circ }\right)+\mathrm{sin}\left({83}^{\circ }\right)\mathrm{sin}\left({23}^{\circ }\right)$

For the following exercises, prove the identity.

$\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(3x\right)\mathrm{cos}x={\mathrm{sin}}^{2}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x$

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4