# 6.3 Inverse trigonometric functions  (Page 7/15)

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Discuss why this statement is incorrect: $\text{\hspace{0.17em}}\mathrm{arccos}\left(\mathrm{cos}\text{\hspace{0.17em}}x\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x.$

Determine whether the following statement is true or false and explain your answer: $\mathrm{arccos}\left(-x\right)=\pi -\mathrm{arccos}\text{\hspace{0.17em}}x.$

True . The angle, $\text{\hspace{0.17em}}{\theta }_{1}\text{\hspace{0.17em}}$ that equals $\text{\hspace{0.17em}}\mathrm{arccos}\left(-x\right)\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}x>0\text{\hspace{0.17em}}$ , will be a second quadrant angle with reference angle, $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ , where $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ equals $\text{\hspace{0.17em}}\mathrm{arccos}x$ , $x>0\text{\hspace{0.17em}}$ . Since $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ is the reference angle for $\text{\hspace{0.17em}}{\theta }_{1}$ , ${\theta }_{2}=\pi -{\theta }_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{arccos}\left(-x\right)\text{\hspace{0.17em}}$ = $\text{\hspace{0.17em}}\pi -\mathrm{arccos}x$ -

## Algebraic

For the following exercises, evaluate the expressions.

${\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)$

${\mathrm{sin}}^{-1}\left(-\frac{1}{2}\right)$

$-\frac{\pi }{6}$

${\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)$

${\mathrm{cos}}^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

$\frac{3\pi }{4}$

${\mathrm{tan}}^{-1}\left(1\right)$

${\mathrm{tan}}^{-1}\left(-\sqrt{3}\right)$

$-\frac{\pi }{3}$

${\mathrm{tan}}^{-1}\left(-1\right)$

${\mathrm{tan}}^{-1}\left(\sqrt{3}\right)$

$\frac{\pi }{3}$

${\mathrm{tan}}^{-1}\left(\frac{-1}{\sqrt{3}}\right)$

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

${\mathrm{cos}}^{-1}\left(-0.4\right)$

1.98

$\mathrm{arcsin}\left(0.23\right)$

$\mathrm{arccos}\left(\frac{3}{5}\right)$

0.93

${\mathrm{cos}}^{-1}\left(0.8\right)$

${\mathrm{tan}}^{-1}\left(6\right)$

1.41

For the following exercises, find the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in the given right triangle. Round answers to the nearest hundredth.

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

${\mathrm{sin}}^{-1}\left(\mathrm{cos}\left(\pi \right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\pi \right)\right)$

0

${\mathrm{cos}}^{-1}\left(\mathrm{sin}\left(\frac{\pi }{3}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{\pi }{3}\right)\right)$

0.71

${\mathrm{sin}}^{-1}\left(\mathrm{cos}\left(\frac{-\pi }{2}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{4\pi }{3}\right)\right)$

-0.71

${\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{5\pi }{6}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{-5\pi }{2}\right)\right)$

$-\frac{\pi }{4}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{4}{5}\right)\right)$

$\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(\frac{3}{5}\right)\right)$

0.8

$\mathrm{sin}\left({\mathrm{tan}}^{-1}\left(\frac{4}{3}\right)\right)$

$\mathrm{cos}\left({\mathrm{tan}}^{-1}\left(\frac{12}{5}\right)\right)$

$\frac{5}{13}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)\right)$

For the following exercises, find the exact value of the expression in terms of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with the help of a reference triangle.

$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(x-1\right)\right)$

$\frac{x-1}{\sqrt{-{x}^{2}+2x}}$

$\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(1-x\right)\right)$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{1}{x}\right)\right)$

$\frac{\sqrt{{x}^{2}-1}}{x}$

$\mathrm{cos}\left({\mathrm{tan}}^{-1}\left(3x-1\right)\right)$

$\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(x+\frac{1}{2}\right)\right)$

$\frac{x+0.5}{\sqrt{-{x}^{2}-x+\frac{3}{4}}}$

## Extensions

For the following exercises, evaluate the expression without using a calculator. Give the exact value.

$\frac{{\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{cos}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\mathrm{cos}}^{-1}\left(1\right)}{{\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{sin}}^{-1}\left(0\right)}$

For the following exercises, find the function if $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=\frac{x}{x+1}.$

$\mathrm{cos}\text{\hspace{0.17em}}t$

$\frac{\sqrt{2x+1}}{x+1}$

$\mathrm{sec}\text{\hspace{0.17em}}t$

$\mathrm{cot}\text{\hspace{0.17em}}t$

$\frac{\sqrt{2x+1}}{x}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{x}{x+1}\right)\right)$

${\mathrm{tan}}^{-1}\left(\frac{x}{\sqrt{2x+1}}\right)$

$t$

## Graphical

Graph $\text{\hspace{0.17em}}y={\mathrm{sin}}^{-1}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

Graph $\text{\hspace{0.17em}}y=\mathrm{arccos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

domain $\text{\hspace{0.17em}}\left[-1,1\right];\text{\hspace{0.17em}}$ range $\text{\hspace{0.17em}}\left[0,\pi \right]\text{\hspace{0.17em}}$

Graph one cycle of $\text{\hspace{0.17em}}y={\mathrm{tan}}^{-1}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

For what value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x={\mathrm{sin}}^{-1}x?\text{\hspace{0.17em}}$ Use a graphing calculator to approximate the answer.

approximately $\text{\hspace{0.17em}}x=0.00\text{\hspace{0.17em}}$

For what value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x={\mathrm{cos}}^{-1}x?\text{\hspace{0.17em}}$ Use a graphing calculator to approximate the answer.

## Real-world applications

Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-ﬂoor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

Suppose you drive 0.6 miles on a road so that the vertical distance changes from 0 to 150 feet. What is the angle of elevation of the road?

An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

Without using a calculator, approximate the value of $\text{\hspace{0.17em}}\mathrm{arctan}\left(10,000\right).\text{\hspace{0.17em}}$ Explain why your answer is reasonable.

A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

#### Questions & Answers

x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake