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Inverse Trigonometric Equation problem

  • Thread starter acen_gr
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  • #1
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EDIT: 1.Homework Statement
The problem asks to simplify this: cos(arcsin x)


Homework Equations


let arc sin x = θ


The Attempt at a Solution


Hoping for some explanation. I really don't have any idea. Thank you so much in advance!
EDIT: I have to edit because I received a warning for not posting for #3 and #2. I originally posted "cos(arcsin x)" for #2 because that was what I know that is relevant to the problem. And I apologize for I didn't know it wasn't enough (I'm new here and sorry if I didn't understand the rules well, I thought it was okay to remove the format instructions and just put what is being asked instead. Quoted above is my previous post #3. As I said there, I really don't have any idea and therefore asks for some (can't put any attempt) When I got the hint from the mentors, I was able to work on this:

The problem asks to simplify this: cos(arcsin x)
Let arcsin x = θ

if arcsin x = θ, then x = sinθ

get the cos of θ from given x/1 and should get for cosθ = √1-x2 (by pythagorean theorem)

the answer is cos(θ) = √1-x2
 
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Answers and Replies

  • #2
Mentallic
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Let [itex]\theta = \arcsin(x)[/itex] so then [itex]x=\sin(\theta)[/itex]. This expression is just simple trig where you first learn that the sine of an angle is OPP/HYP in a right triangle right? So do just that. Create the right-triangle with an angle of [itex]\theta[/itex] such that [itex]\sin(\theta)=x[/itex] (which is equivalent to x/1).
 
  • #3
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Let [itex]\theta = \arcsin(x)[/itex] so then [itex]x=\sin(\theta)[/itex]. This expression is just simple trig where you first learn that the sine of an angle is OPP/HYP in a right triangle right? So do just that. Create the right-triangle with an angle of [itex]\theta[/itex] such that [itex]\sin(\theta)=x[/itex] (which is equivalent to x/1).
So what should be my final answer? Is it Cos θ? Sorry if I ask too many. :(

EDIT: Ok sir. Nevermind this. I already got the answer. Thank you so much! :)
 
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  • #4
HallsofIvy
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Good! I am glad you were able to get the answer- I hope you were able to work it out for yourself from the hint acen gr gave you.

For others who might wonder: Yes, "the answer is [itex]cos(\theta)[/itex]" but that is not the end of it. As acen gr suggested, "Create the right-triangle with an angle of θ such that sin(θ)=x (which is equivalent to x/1)." Then we have a right triangle with "opposite side" of length x and "hypotenuse" of length 1. By the Pythagorean theorem, the "near side" has length [itex]\sqrt{1- x^2}[/itex]. Since cosine is "near side over hypotenuse", [itex]cos(\theta)= cos(cos^{-1}(x))= \frac{(\sqrt{1- x^2}}{1}= \sqrt{1- x^2}[/itex]

You could also do it this way: for any angle [itex]\theta[/itex], [itex]sin^2(\theta)+ cos^2(\theta)= 1[/itex] so that [itex]cos(\theta)= \pm\sqrt{1- sin^2(\theta)}[/itex]. That is, [itex]cos(sin^{-1}(x))= \pm\sqrt{1- sin^2(sin^{-1}(x))}= \pm\sqrt{1- x^2}[/itex]. Because the standard domain of [itex]sin^{-1}[/itex] is [itex]-\pi/2[/itex] to [itex]\pi/2[/itex], the sign is the same as the sign of x.

(The sign did not come up in the previous paragraph because being able to draw a triangle with side "x" assumes x is positive.)
 
  • #5
Mentallic
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HallsofIvy, I think you're getting a little mixed up with who is helping who :tongue:
 
  • #6
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Good! I am glad you were able to get the answer- I hope you were able to work it out for yourself from the hint acen gr gave you.

For others who might wonder: Yes, "the answer is [itex]cos(\theta)[/itex]" but that is not the end of it. As acen gr suggested, "Create the right-triangle with an angle of θ such that sin(θ)=x (which is equivalent to x/1)." Then we have a right triangle with "opposite side" of length x and "hypotenuse" of length 1. By the Pythagorean theorem, the "near side" has length [itex]\sqrt{1- x^2}[/itex]. Since cosine is "near side over hypotenuse", [itex]cos(\theta)= cos(cos^{-1}(x))= \frac{(\sqrt{1- x^2}}{1}= \sqrt{1- x^2}[/itex]

You could also do it this way: for any angle [itex]\theta[/itex], [itex]sin^2(\theta)+ cos^2(\theta)= 1[/itex] so that [itex]cos(\theta)= \pm\sqrt{1- sin^2(\theta)}[/itex]. That is, [itex]cos(sin^{-1}(x))= \pm\sqrt{1- sin^2(sin^{-1}(x))}= \pm\sqrt{1- x^2}[/itex]. Because the standard domain of [itex]sin^{-1}[/itex] is [itex]-\pi/2[/itex] to [itex]\pi/2[/itex], the sign is the same as the sign of x.

(The sign did not come up in the previous paragraph because being able to draw a triangle with side "x" assumes x is positive.)
Sir, so the final answer should be Cosθ = [itex]\sqrt{1- x^2}[/itex]? Am I right?

EDIT: Anyway sir, I am the one who was helped by Mentallic :)
EDIT: How do I make square root symbol here?
 
  • #7
HallsofIvy
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HallsofIvy, I think you're getting a little mixed up with who is helping who :tongue:
One of these days, I really need to learn to read!
 
  • #8
HallsofIvy
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Sir, so the final answer should be Cosθ = [itex]\sqrt{1- x^2}[/itex]? Am I right?
Assuming that x is positive, yes. Otherwise [itex]cos(sin(\theta))= \pm\sqrt{1- x^2}[/itex].

EDIT: Anyway sir, I am the one who was helped by Mentallic :)
EDIT: How do I make square root symbol here?
There is a LaTeX tutorial at https://www.physicsforums.com/showthread.php?p=3977517&posted=1#post3977517.

For just a single square root symbol, if you click on the "Go Advanced", you will see a table of "Quick Symbols" and there is a "√" there- notice that you had better use ( ) with that!
 
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  • #9
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Assuming that x is positive, yes. Otherwise [itex]cos(sin(\theta))= \pm\sqrt{1- x^2}[/itex].
Wee! Thanks mentor! :) I love this site!
 
  • #10
Mentallic
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Sir, so the final answer should be Cosθ = [itex]\sqrt{1- x^2}[/itex]? Am I right?
Yes that's correct.

How do I make square root symbol here?
You already have! Using tex tags as you've done.

One of these days, I really need to learn to read!
Create a homework thread and I'm sure some of us would be delighted to help you :wink:
 
  • #11
Mentallic
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Assuming that x is positive, yes. Otherwise [itex]cos(sin(\theta))= \pm\sqrt{1- x^2}[/itex].
Just to clarify, we're talking about [tex]\cos(\sin^{-1}\theta)[/tex] and not [tex]\cos(\sin\theta)[/tex]

Also, the range of [itex]\sin^{-1}\theta[/itex] is [itex][-\pi/2, \pi/2][/itex] and thus the range of [itex]\cos(\sin^{-1}\theta)[/itex] is [itex][0,1][/itex] so the answer is only [itex]\sqrt{1-x^2}[/itex]
 
  • #12
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Just to clarify, we're talking about [tex]\cos(\sin^{-1}\theta)[/tex] and not [tex]\cos(\sin\theta)[/tex]

Also, the range of [itex]\sin^{-1}\theta[/itex] is [itex][-\pi/2, \pi/2][/itex] and thus the range of [itex]\cos(\sin^{-1}\theta)[/itex] is [itex][0,1][/itex] so the answer is only [itex]\sqrt{1-x^2}[/itex]
Oh I see. No wonder why the actual question here in my notes asks for an exact answer;) Thanks Metallic! :D
 

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