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Trigonometric Equations

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  • #1
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Homework Statement


Find the solution set of the equation below for the interval 0° ≤ x ≤ 360°:

arccos(x) - arcsin(1-x) = 90°

Homework Equations


all given above.


The Attempt at a Solution


i tried to consider (1-x) as "complementary" but I'm not really sure about that and I would like to know your opinion if I should go for it? And I'm kinda confused why the RHS is in degrees. Is it because the problem is dealing with inverse trig equations (which the answers are supposed to be angles)? Thanks in advance.
 

Answers and Replies

  • #2
Simon Bridge
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Homework Statement


Find the solution set of the equation below for the interval 0° ≤ x ≤ 360°:
That doesn't make sense - surely it is -1 ≤ x ≤ 1 (angle is the output of inverse trig functions).

arccos(x) - arcsin(1-x) = 90°
Hmmm ... well, [tex]\arccos(x)+\arcsin(x)=90^\circ[/tex]

Homework Equations


all given above.
We-ell, there are a whole lot of trig identities that may be useful too.

The Attempt at a Solution


i tried to consider (1-x) as "complementary" but I'm not really sure about that and I would like to know your opinion if I should go for it? And I'm kinda confused why the RHS is in degrees. Is it because the problem is dealing with inverse trig equations (which the answers are supposed to be angles)? Thanks in advance.
... the RHS is the result of the sum of two angles, so, naturally it is in degrees.

I'd have been tempted to take the cosine of both sides and use the sum-to-product relations with the trig-inverse-trig relations to get some f(x)=0 ... then it's a matter of finding the roots of f.

But your idea is cool too ... if you can show that arcsin(1-x) = -arcsin(x) you'd be made.

eg. if x=1, then 1-x=0, arcsin(1)=90, arcsin(0)=0 or 180.
So - not generally true. But maybe it is true for some value of x?

Of course, plotting the function y=arccos(x)-arcsin(1-x) could give a few clues.
 
  • #3
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And I'm kinda confused why the RHS is in degrees.
No particular reason, I think. You can write them in radians too if you wish. So you could as well write

[tex]arccos(x)-arcsin(1-x)=\frac{\pi}{2}[/tex]

It's the same thing, although I prefer to write them like this.

Anyway, try to write the equation as

[tex]arccos(x)=\frac{\pi}{2}+arcsin(1-x)[/tex]

and now take the cosine of both sides.
 
  • #4
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[tex]arccos(x)=\frac{\pi}{2}+arcsin(1-x)[/tex]

and now take the cosine of both sides.

Hi, micromass. Thanks for posting a reply.
I'm working on it now and I just want to confirm if I'm doing this right:

Taking cosine of both sides will give

[tex]0 = \cos\frac{\pi}{2} + \cos[\arcsin(1-x)][/tex]

Question: In cos[arcsin(1-x)], is it right to use "substitution" method where I let, say θ, equal to arcsin(1-x), then find for cosθ afterwards?

It's something like: cos[arcsin(1-x)]

let θ = arcsin(1-x)

sinθ = 1-x then is it right to find for cosθ through this?

Thanks!

EDIT: I guess this is wrong T_T Please tell me and sorry coz I'm still a learner T_T
EDIT: I'm having trouble with latex again.
 
Last edited:
  • #5
Mentallic
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Taking cosine of both sides will give

[tex]0 = \cos\frac{\pi}{2} + \cos[\arcsin(1-x)][/tex]
[tex]\cos(A+B)\neq \cos(A)+\cos(B)[/tex]

Remember it's

[tex]\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)[/tex]

Question: In cos[arcsin(1-x)], is it right to use "substitution" method where I let, say θ, equal to arcsin(1-x) then find for cosθ afterwards?

It's something like: cos[arcsin(1-x)]

let θ = arcsin(1-x)

sinθ = 1-x then is it right to find for cosθ through this?

Thanks!

EDIT: I guess this is wrong T_T Please tell me and sorry coz I'm still a learner T_T
Yes you're on the right track. So if [itex]\sin(\theta) = 1-x[/itex] then draw up a right-angled triangle with one angle being [itex]\theta[/itex] and then label the sides in such a way such that [itex]\sin(\theta)=1-x[/itex]
Then, since we are actually looking for the value of [itex]\cos(\arcsin(1-x))\equiv \cos(\theta)[/itex] you'll need to find the value of the other side in the right-triangle.
 
  • #6
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[tex]\cos(A+B)\neq \cos(A)+\cos(B)[/tex]

Remember it's

[tex]\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)[/tex]
Oh yes!! Why do I always forget that. Thanks a lot! Ok, I'm gonna work on it again. Thank you for all your replies! :)
 
  • #7
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Alright. Summing up everyone's ideas, I got this solution:

Anyway, try to write the equation as

[tex]arccos(x)=\frac{\pi}{2}+arcsin(1-x)[/tex]

and now take the cosine of both sides.
[tex] cos[arccos(x) = \frac{\pi}{2} + arcsin(1-x)] [/tex]
[tex] 0 = cos[\frac{\pi}{2} + arcsin(1-x)] [/tex]


take the cosine of both sides and use the sum-to-product relations with the trig-inverse-trig relations to get some f(x)=0 ... then it's a matter of finding the roots of f.
Remember it's
[tex]\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)[/tex]
[tex]0 = cos\frac{\pi}{2}cos[arcsin(1-x)] - sin\frac{\pi}{2}cos[arcsin(1-x)[/tex]
[tex]0 = cos[arcsin(1-x)][/tex]

Then, since we are actually looking for the value of [itex]\cos(\arcsin(1-x))\equiv \cos(\theta)[/itex] you'll need to find the value of the other side in the right-triangle.
[tex] let θ = arcsin(1-x)[/tex] ⇔ sinθ = 1-x

for angle θ: y = 1-x, x = [itex]\sqrt{2x-x^2}[/itex], r = 1

∴ cosθ = [itex]\sqrt{2x-x^2}[/itex]

[itex] 0 = -cos(θ) [/itex]
[itex]0 = -(\sqrt{2x-x^2})[/itex]
[itex]0 = \sqrt{2x-x^2}[/itex]
[tex]0 = 2x-x^2[/tex]
[tex]0 = (2-x)x[/tex]

[tex]x = 2 , x = 0[/tex]

surely it is -1 ≤ x ≤ 1 (angle is the output of inverse trig functions).
so x = 0.

Hope it's right now.
 
  • #8
Simon Bridge
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note: arccos is an inverse cosine so:[tex]\cos(\arccos(x))=x[/tex]

Also, you'll find there is an identity for [itex]\cos(\frac{\pi}{2}+\theta)[/itex] that will be useful for simplifying things.
You'll still get it the long way around though.
 
  • #9
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note: arccos is an inverse cosine so:[tex]\cos(\arccos(x))=x[/tex]

Also, you'll find there is an identity for [itex]\cos(\frac{\pi}{2}+\theta)[/itex] that will be useful for simplifying things.
You'll still get it the long way around though.
oh yeaahh. i forgot T____T
how am i going to pass my test if i always keep on making mistakes.
my test is already tomorrow, and i got that problem above from a sample problem of what might appear as question for tomorrow's exam.
i think i would fail again :(
how did you guys became so gifted in math?
 
  • #10
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it's already late here and i'm still not yet through studying for exam :( please check my new solution. I still hope I could get it:


[itex] x = -cos(θ) [/itex]
[itex] x= -\sqrt{2x-x^2}[/itex]
[itex] -x = \sqrt{2x-x^2}[/itex]
[tex] -x^2 = 2x-x^2[/tex]
[tex] 0 - 2x = 0[/tex]
[tex] x = 0 [/tex]
 
  • #11
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it's already late here and i'm still not yet through studying for exam :( please check my new solution. I still hope I could get it:


[itex] x = -cos(θ) [/itex]
[itex] x= -\sqrt{2x-x^2}[/itex]
[itex] -x = \sqrt{2x-x^2}[/itex]
[tex] -x^2 = 2x-x^2[/tex]
When you square -x, you get x2, not -x2.
You could have squared both sides of the 2nd equation, above.
[tex] 0 - 2x = 0[/tex]
[tex] x = 0 [/tex]
 
  • #12
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how about this:

[itex] x = -cos(θ) [/itex]
[itex] x= -\sqrt{2x-x^2}[/itex]
[tex] x^2 = 2x-x^2[/tex]
[tex] 2x^2 - 2x = 0[/tex]
[tex] x^2 - x = 0 [/tex]

x = 1, x = 0
 
  • #13
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That's better, but you're still not done. When you square both sides of an equation, there is the possibility that you are introducing extraneous solutions.

Check both solutions to see if they satisfy the equation you squared: x = -√(2x - x2).
 
  • #14
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That's better, but you're still not done. When you square both sides of an equation, there is the possibility that you are introducing extraneous solutions.

Check both solutions to see if they satisfy the equation you squared: x = -√(2x - x2).
The answer is x = 0.
Because if x = 1, then

[tex]x = -\sqrt{2x - x^2}[/tex]
[tex](1) = -\sqrt{2(1) - 1^2}[/tex]
[tex] 1 = -\sqrt{1}[/tex]
[tex] 1 ≠ -1 [/tex]

if x = 0

[tex]0 = -\sqrt{0 - 0^2}[/tex]
[tex](0) = -\sqrt{0}[/tex]
[tex] 0 = 0 [/tex]

Am I right sir?
 
  • #15
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That is the correct solution of the equation x = -√(2x - x2), but it's not a solution of your original equation. I haven't checked all of your work, but there must be an error somewhere.
 
  • #16
Simon Bridge
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Yeh - not seeing where [itex]x=-\sqrt{2x-x^2}[/itex] comes from.

[edit] no I got it - it's from the combined trig and inverse-trig identity for cos(arcsin(x))
But that doesn't account for the extra [itex]\frac{\pi}{2}[/itex] in the RHS.

He seems to have gone from:
[tex]\arccos(x) = \frac{\pi}{2} + \arcsin(1-x)[/tex]... to
[tex]x = \cos \big [ \arcsin(1-x) \big ] = \sqrt{1-(1-x)^2} [/tex] (don't know where the minus sign came from either.)

I detect tiredness errors.

hint: [itex]\cos(\frac{\pi}{2})=0[/itex]
 
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  • #17
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[tex]x = \cos \big [ \arcsin(1-x) \big ] = \sqrt{1-(1-x)^2} [/tex] (don't know where the minus sign came from either.)
That was from extracting (1-x)^2
 
  • #18
Mentallic
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[tex]\sqrt{x^2-2x}\neq - \sqrt{2x-x^2}[/tex]

Remember the rule that [itex]\sqrt{ab}=\sqrt{a}\sqrt{b}[/itex] so this would imply that [itex]\sqrt{x^2-2x}=\sqrt{-(2x-x^2)}=\sqrt{-1}\sqrt{2x-x^2}[/itex] and the square root of -1 is not -1, it's imaginary. But the rule only applies for positive a and b values, so even then we can't do that.

You need to leave it as it is.
 
  • #19
Simon Bridge
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That was from extracting (1-x)^2
The minus sign outside the surd? Mentallic has described why that should not come from anything inside the surd.

go back to:
[tex]x=\cos\left [ \frac{\pi}{2} + \arcsin(1-x) \right ] [/tex]... put [itex]\theta=\arcsin(1-x)[/itex] and simplify: what does [itex]\cos(\frac{\pi}{2}+\theta)[/itex] turn into?
 
  • #20
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The minus sign outside the surd? Mentallic has described why that should not come from anything inside the surd.

go back to:
[tex]x=\cos\left [ \frac{\pi}{2} + \arcsin(1-x) \right ] [/tex]... put [itex]\theta=\arcsin(1-x)[/itex] and simplify: what does [itex]\cos(\frac{\pi}{2}+\theta)[/itex] turn into?
this?

let θ = arcsin(1-x)

[tex]x=\cos\left [ \frac{\pi}{2} + \arcsin(1-x) \right ][/tex]

[tex]x=\cos\left [ \frac{\pi}{2} + θ \right ][/tex]

[tex]x=cos\frac{\pi}{2}cosθ - sinθsin\frac{\pi}{2}[/tex]

x = -sinθ
 
  • #21
Simon Bridge
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... wasn't that fun?
Now something should naturally occur to you ... you know what sinθ is!

BTW: I have a problem with this and I'm hoping someone can see the flaw in the reasoning.
I have a headslap moment coming ... maybe if I sleep on it.
 
  • #22
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@simon bridge, yea. and i'm still struggling because we just had this question in our exam today and our teacher afterwards told the answer was x = -1/2. is that right?
 
  • #23
Simon Bridge
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BTW: what was wrong with using the identity in post #2?
 
  • #24
Simon Bridge
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@simon bridge, yea. and i'm still struggling because we just had this question in our exam today and our teacher afterwards told the answer was x = -1/2. is that right?
well ... the second term on the LHS of the original problem in post #1 is arcsin(1-x).

if x=-0.5, then that term becomes arcsin(1.5) = undefined.

so how can that be the solution to the problem stated in post #1?

However: x=0.5 is the solution.
 
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  • #25
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yea. that's how we argued with him. it doesn't seem right and he doesn't seem to know the right answer too. now i'm so confused.

btw, is this the identity you were referring to in reply#23?

[tex]\arccos(x)+\arcsin(x)=90^\circ[/tex]

UPDATE: I think x could be equal to 1/2. What do you think?
 

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