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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 $\text{\hspace{0.17em}}0.1\text{cm}\text{.}$
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 $\text{\hspace{0.17em}}0.1\text{cm}\text{.}$
Spring 2 comes to rest first after 8.0 seconds.
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 $\text{}{60}^{\circ},$ 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 |
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 $
${\mathrm{sin}}^{-1}\left(\frac{1}{4}\right),\pi -{\mathrm{sin}}^{-1}\left(\frac{1}{4}\right)$
$1-2\text{\hspace{0.17em}}\mathrm{tan}(\omega )={\mathrm{tan}}^{2}(\omega )$
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
${\mathrm{tan}}^{3}x\text{\hspace{0.17em}}{\mathrm{csc}}^{2}x\text{\hspace{0.17em}}{\mathrm{cot}}^{2}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x=1$
For the following exercises, find the exact value.
$\mathrm{cos}\left(\frac{25\pi}{12}\right)$
$\mathrm{sin}\left({70}^{\circ}\right)\mathrm{cos}\left({25}^{\circ}\right)-\mathrm{cos}\left({70}^{\circ}\right)\mathrm{sin}\left({25}^{\circ}\right)$
$\frac{\sqrt{2}}{2}$
$\mathrm{cos}\left({83}^{\circ}\right)\mathrm{cos}\left({23}^{\circ}\right)+\mathrm{sin}\left({83}^{\circ}\right)\mathrm{sin}\left({23}^{\circ}\right)$
For the following exercises, prove the identity.
$\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(3x\right)\mathrm{cos}x={\mathrm{sin}}^{2}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x$
$\begin{array}{l}\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(3x\right)\mathrm{cos}x=\mathrm{cos}\left(2x+2x\right)-\mathrm{cos}\left(x+2x\right)\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ \text{}=\mathrm{cos}\left(2x\right)\mathrm{cos}\left(2x\right)-\mathrm{sin}\left(2x\right)\mathrm{sin}\left(2x\right)-\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x+\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ \text{}={\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)}^{2}-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+\mathrm{sin}\text{\hspace{0.17em}}x\left(2\right)\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\hfill \\ \text{}={\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)}^{2}-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)+2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\hfill \\ \text{}={\mathrm{cos}}^{4}x-2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+{\mathrm{sin}}^{4}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x-{\mathrm{cos}}^{4}x+{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\hfill \\ \text{}={\mathrm{sin}}^{4}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\hfill \\ \text{}={\mathrm{sin}}^{2}x\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x\text{\hspace{0.17em}}{\mathrm{sin}}^{2}x\hfill \\ \text{}={\mathrm{sin}}^{2}x-4\text{\hspace{0.17em}}{\mathrm{cos}}^{2}x{\mathrm{sin}}^{2}x\hfill \end{array}$
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