<< Chapter < Page Chapter >> Page >

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  cm .

Got questions? Get instant answers now!

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  cm .

Spring 2 comes to rest first after 8.0 seconds.

Got questions? Get instant answers now!

Extensions

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

Got questions? Get instant answers now!

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

500 miles, at 90

Got questions? Get instant answers now!

For the following exercises, find a function of the form y = a b x + c sin ( π 2 x ) that fits the given data.

x 0 1 2 3
y 6 34 150 746

y = 6 ( 5 ) x + 4 sin ( π 2 x )

Got questions? Get instant answers now!

For the following exercises, find a function of the form y = a b x cos ( π 2 x ) + c that fits the given data.

x 0 1 2 3
y 11 3 1 3

y = 8 ( 1 2 ) x cos ( π 2 x ) + 3

Got questions? Get instant answers now!

Chapter review exercises

Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval [ 0 , 2 π ) .

csc 2 t = 3

sin 1 ( 3 3 ) , π sin 1 ( 3 3 ) , π + sin 1 ( 3 3 ) , 2 π sin 1 ( 3 3 )

Got questions? Get instant answers now!

2 sin θ = 1

7 π 6 , 11 π 6

Got questions? Get instant answers now!

tan x sin x + sin ( x ) = 0

Got questions? Get instant answers now!

9 sin ω 2 = 4 sin 2 ω

sin 1 ( 1 4 ) , π sin 1 ( 1 4 )

Got questions? Get instant answers now!

1 2 tan ( ω ) = tan 2 ( ω )

Got questions? Get instant answers now!

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

sec x cos x + cos x 1 sec x

1

Got questions? Get instant answers now!

sin 3 x + cos 2 x sin x

Got questions? Get instant answers now!

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

sin 2 x + sec 2 x 1 = ( 1 cos 2 x ) ( 1 + cos 2 x ) cos 2 x

Yes

Got questions? Get instant answers now!

tan 3 x csc 2 x cot 2 x cos x sin x = 1

Got questions? Get instant answers now!

Sum and Difference Identities

For the following exercises, find the exact value.

tan ( 7 π 12 )

2 3

Got questions? Get instant answers now!

sin ( 70 ) cos ( 25 ) cos ( 70 ) sin ( 25 )

2 2

Got questions? Get instant answers now!

cos ( 83 ) cos ( 23 ) + sin ( 83 ) sin ( 23 )

Got questions? Get instant answers now!

For the following exercises, prove the identity.

cos ( 4 x ) cos ( 3 x ) cos x = sin 2 x 4 cos 2 x sin 2 x

cos ( 4 x ) cos ( 3 x ) cos x = cos ( 2 x + 2 x ) cos ( x + 2 x ) cos x                                    = cos ( 2 x ) cos ( 2 x ) sin ( 2 x ) sin ( 2 x ) cos x cos ( 2 x ) cos x + sin x sin ( 2 x ) cos x                                    = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x ( cos 2 x sin 2 x ) + sin x ( 2 ) sin x cos x cos x                                    = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x ( cos 2 x sin 2 x ) + 2 sin 2 x cos 2 x                                    = cos 4 x 2 cos 2 x sin 2 x + sin 4 x 4 cos 2 x sin 2 x cos 4 x + cos 2 x sin 2 x + 2 sin 2 x cos 2 x                                    = sin 4 x 4 cos 2 x sin 2 x + cos 2 x sin 2 x                                    = sin 2 x ( sin 2 x + cos 2 x ) 4 cos 2 x sin 2 x                                    = sin 2 x 4 cos 2 x sin 2 x

Got questions? Get instant answers now!
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask