7.6 Modeling with trigonometric equations  (Page 10/14)

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 $0.1\text{ cm}\text{.}$

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 $0.1\text{ cm}\text{.}$

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.

A plane flies 2 hours at 200 mph at a bearing of ${60}^{\circ },$ then continues to fly for 1.5 hours at the same speed, this time at a bearing of ${150}^{\circ }.$ Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.

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

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

$2\sin \theta =-1$

$\tan x\sin x+\sin \left(-x\right)=0$

$9\sin \omega -2=4{\sin }^2\omega$

$1-2\tan (\omega )={\tan }^2(\omega )$

$\sec x\cos x+\cos x-\frac{1}{\sec x}$

${\sin }^{3}x+{\cos }^{2}x\sin x$

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

${\tan }^{3}x{\csc }^{2}x{\cot }^{2}x\cos x\sin x=1$

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

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

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

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

$\cos \left(4x\right)-\cos \left(3x\right)\cos x={\sin }^{2}x-4{\cos }^{2}x{\sin }^{2}x$