# 8.3 Inverse trigonometric functions  (Page 8/15)

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The line $\text{\hspace{0.17em}}y=\frac{3}{5}x\text{\hspace{0.17em}}$ 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?

The line $\text{\hspace{0.17em}}y=\frac{-3}{7}x\text{\hspace{0.17em}}$ 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?

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.)

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.

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?

## 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.

$f\left(x\right)=-3\mathrm{cos}\text{\hspace{0.17em}}x+3$

amplitude: 3; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=3;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=\frac{1}{4}\mathrm{sin}\text{\hspace{0.17em}}x$

$f\left(x\right)=3\mathrm{cos}\left(x+\frac{\pi }{6}\right)$

amplitude: 3; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=-2\mathrm{sin}\left(x-\frac{2\pi }{3}\right)$

$f\left(x\right)=3\mathrm{sin}\left(x-\frac{\pi }{4}\right)-4$

amplitude: 3; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=-4;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=2\left(\mathrm{cos}\left(x-\frac{4\pi }{3}\right)+1\right)$

$f\left(x\right)=6\mathrm{sin}\left(3x-\frac{\pi }{6}\right)-1$

amplitude: 6; period: $\text{\hspace{0.17em}}\frac{2\pi }{3};\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=-1;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=-100\mathrm{sin}\left(50x-20\right)$

## 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.

$f\left(x\right)=\mathrm{tan}\text{\hspace{0.17em}}x-4$

stretching factor: none; period: midline: asymptotes: where is an integer

$f\left(x\right)=2\mathrm{tan}\left(x-\frac{\pi }{6}\right)$

$f\left(x\right)=-3\mathrm{tan}\left(4x\right)-2$

stretching factor: 3; period: midline: asymptotes: $x=\frac{\pi }{8}+\frac{\pi }{4}k,$ where is an integer

$f\left(x\right)=0.2\mathrm{cos}\left(0.1x\right)+0.3$

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

$f\left(x\right)=\frac{1}{3}\mathrm{sec}\text{\hspace{0.17em}}x$

amplitude: none; period: $2\pi ;$ no phase shift; asymptotes: where is an odd integer

$f\left(x\right)=3\mathrm{cot}\text{\hspace{0.17em}}x$

$f\left(x\right)=4\mathrm{csc}\left(5x\right)$

amplitude: none; period: no phase shift; asymptotes: where is an integer

$f\left(x\right)=8\mathrm{sec}\left(\frac{1}{4}x\right)$

$f\left(x\right)=\frac{2}{3}\mathrm{csc}\left(\frac{1}{2}x\right)$

amplitude: none; period: no phase shift; asymptotes: where is an integer

$f\left(x\right)=-\mathrm{csc}\left(2x+\pi \right)$

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: $\text{\hspace{0.17em}}y=12,000+8,000\mathrm{sin}\left(0.628x\right),\text{\hspace{0.17em}}$ 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

Graph the function on the domain of $\text{\hspace{0.17em}}\left[0,40\right]$ .

Cos45/sec30+cosec30=
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1