# 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.

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

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