<< Chapter < Page Chapter >> Page >

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.)

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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].
Got questions? Get instant answers now!

f ( x ) = 2 sin ( x 2 π 3 )

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

f ( x ) = 2 ( cos ( x 4 π 3 ) + 1 )

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

f ( x ) = 100 sin ( 50 x 20 )

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

f ( x ) = 2 tan ( x π 6 )

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

f ( x ) = 0.2 cos ( 0.1 x ) + 0.3

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

f ( x ) = 8 sec ( 1 4 x )

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

f ( x ) = csc ( 2 x + π )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask