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The line y = 3 5 x passes through the origin in the x , y -plane. What is the measure of the angle that the line makes with the positive x -axis?

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The line y = 3 7 x passes through the origin in the x , y -plane. What is the measure of the angle that the line makes with the negative x -axis?

0.405 radians

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What percentage grade should a road have if the angle of elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100-foot horizontal distance. For example a 5% grade means that the road rises 5 feet for every 100 feet of horizontal distance.)

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A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder's angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?

No. The angle the ladder makes with the horizontal is 60 degrees.

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Suppose a 15-foot ladder leans against the side of a house so that the angle of elevation of the ladder is 42 degrees. How far is the foot of the ladder from the side of the house?

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Chapter review exercises

Graphs of the Sine and Cosine Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

f ( x ) = 3 cos x + 3

amplitude: 3; period: 2 π ; midline: y = 3 ; no asymptotes

A graph of two periods of a function with a cosine parent function. The graph has a range of [0,6] graphed over -2pi to 2pi. Maximums as -pi and pi.
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f ( x ) = 3 cos ( x + π 6 )

amplitude: 3; period: 2 π ; midline: y = 0 ; no asymptotes

A graph of four periods of a function with a cosine parent function. Graphed from -4pi to 4pi. Range is [-3,3].
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f ( x ) = 2 sin ( x 2 π 3 )

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f ( x ) = 3 sin ( x π 4 ) 4

amplitude: 3; period: 2 π ; midline: y = 4 ; no asymptotes

A graph of two periods of a sinusoidal function. Range is [-7,-1]. Maximums at -5pi/4 and 3pi/4.
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f ( x ) = 2 ( cos ( x 4 π 3 ) + 1 )

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f ( x ) = 6 sin ( 3 x π 6 ) 1

amplitude: 6; period: 2 π 3 ; midline: y = 1 ; no asymptotes

A sinusoidal graph over two periods. Range is [-7,5], amplitude is 6, and period is 2pi/3.
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f ( x ) = 100 sin ( 50 x 20 )

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Graphs of the Other Trigonometric Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

f ( x ) = tan x 4

stretching factor: none; period:   π ;   midline:   y = 4 ;   asymptotes:   x = π 2 + π k , where   k   is an integer

A graph of a tangent function over two periods. Graphed from -pi to pi, with asymptotes at -pi/2 and pi/2.
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f ( x ) = 2 tan ( x π 6 )

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f ( x ) = 3 tan ( 4 x ) 2

stretching factor: 3; period:   π 4 ;   midline:   y = 2 ;   asymptotes: x = π 8 + π 4 k , where   k   is an integer

A graph of a tangent function over two periods. Asymptotes at -pi/8 and pi/8. Period of pi/4. Midline at y=-2.
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f ( x ) = 0.2 cos ( 0.1 x ) + 0.3

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For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.

f ( x ) = 1 3 sec x

amplitude: none; period: 2 π ; no phase shift; asymptotes:   x = π 2 k , where   k   is an odd integer

A graph of two periods of a secant function. Period of 2 pi, graphed from -2pi to 2pi. Asymptotes at -3pi/2, -pi/2, pi/2, and 3pi/2.
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f ( x ) = 4 csc ( 5 x )

amplitude: none; period:   2 π 5 ;   no phase shift; asymptotes:   x = π 5 k , where   k   is an integer

A graph of a cosecant functionover two and a half periods. Graphed from -pi to pi, period of 2pi/5.
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f ( x ) = 8 sec ( 1 4 x )

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f ( x ) = 2 3 csc ( 1 2 x )

amplitude: none; period:   4 π ;   no phase shift; asymptotes:   x = 2 π k , where   k   is an integer

A graph of two periods of a cosecant function. Graphed from -4pi to 4pi. Asymptotes at multiples of 2pi. Period of 4pi.
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f ( x ) = csc ( 2 x + π )

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For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: y = 12 , 000 + 8 , 000 sin ( 0.628 x ), where the domain is the years since 1980 and the range is the population of the city.

What is the largest and smallest population the city may have?

largest: 20,000; smallest: 4,000

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Graph the function on the domain of [ 0 , 40 ] .

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Questions & Answers

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
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Practice Key Terms 6

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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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