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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 0.1  cm .

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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.

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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.

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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

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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 )

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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

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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 )

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2 sin θ = 1

7 π 6 , 11 π 6

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tan x sin x + sin ( x ) = 0

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9 sin ω 2 = 4 sin 2 ω

sin 1 ( 1 4 ) , π sin 1 ( 1 4 )

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1 2 tan ( ω ) = tan 2 ( ω )

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

sec x cos x + cos x 1 sec x

1

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sin 3 x + cos 2 x sin x

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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

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

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Sum and Difference Identities

For the following exercises, find the exact value.

tan ( 7 π 12 )

2 3

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

2 2

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

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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

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Questions & Answers

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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