Trigonometry problem

  • Thread starter terryds
  • Start date
  • #1
392
13

Homework Statement



If ##cos^3x+\frac{1}{cos^3x} = 0##, then sin 2x equals ...

A. -1
B. -√3
C. -√2
D. 1
E. 2

Homework Equations


Basic trigonometry

The Attempt at a Solution


[/B]
Let y = cos^3 x

y+1/y = 0
y^2 + 1 = 0
y = √(-1)

Then, I don't know what to do.. Help please..

 

Answers and Replies

  • #2
member 587159

Homework Statement



If ##cos^3x+\frac{1}{cos^3x} = 0##, then sin 2x equals ...

A. -1
B. -√3
C. -√2
D. 1
E. 2

Homework Equations


Basic trigonometry

The Attempt at a Solution


[/B]
Let y = cos^3 x

y+1/y = 0
y^2 + 1 = 0
y = √(-1)

Then, I don't know what to do.. Help please..
I think you can solve this one for cosx. Make the expression into one fraction and remember a/b = 0 <=> a = 0 when b =/= 0.
 
  • #3
392
13
I think you can solve this one for cosx. Make the expression into one fraction and remember a/b = 0 <=> a = 0 when b =/= 0.

##\frac{cos^6x+1}{cos^3x}=0##
##cos^6x = -1##
##cos x =\sqrt[6]{-1}##
##sin 2x = 2 sin x cos x \\
sin 2x = 2 \sqrt{1-\sqrt[3]{-1}}\sqrt[6]{-1}##

How to solve it????
 
  • #4
member 587159
Well, in the context of real numbers, there does not seem to be an x that satisfies the equation.
 
  • #5
392
13
Well, in the context of real numbers, there does not seem to be an x that satisfies the equation.
So, no option satisfies??
 
  • #6
blue_leaf77
Science Advisor
Homework Helper
2,629
784
Then ##x## must be complex. From the original equation, out of the two possibilities you can choose ##\cos^3 x = i = e^{i\pi/2}##. From this, find ##\cos^2 x## and ##\cos x##, then ##\sin x##.
 
  • #7
392
13
Then ##x## must be complex. From the original equation, out of the two possibilities you can choose ##\cos^3 x = i = e^{i\pi/2}##. From this, find ##\cos^2 x## and ##\cos x##, then ##\sin x##.
It seems a bit complicated to take the cubic root from exponential form.
Is there any easier way??
During test, calculator is not allowed.. So I try not using calculator since this is my preparation
 
  • #8
blue_leaf77
Science Advisor
Homework Helper
2,629
784
No, it's fairly easy actually.
From ##\cos^3 x = e^{i\pi/2}##, find ##\cos x ## in terms of complex exponential.
 
  • #9
392
13
No, it's fairly easy actually.
From ##\cos^3 x = e^{i\pi/2}##, find ##\cos x ## in terms of complex exponential.
##\cos x = e^{i\pi/6} \\
sin x = \sqrt{1-e^{i\pi/3}} \\
sin 2x = 2 sin\ x\ cos\ x\ = 2\sqrt{1-e^{i\pi/3}} e^{i\pi/6}##

Now, it's complicated.
How to solve it?
 
  • #10
blue_leaf77
Science Advisor
Homework Helper
2,629
784
Not that complicated if you have put some more creativity and trials, now in
$$
sin x = \sqrt{1-e^{i\pi/3}} \\
$$
express ##1-e^{i\pi/3}## in ##x+iy## form.
 
  • #11
392
13
Not that complicated if you have put some more creativity and trials, now in
$$
sin x = \sqrt{1-e^{i\pi/3}} \\
$$
express ##1-e^{i\pi/3}## in ##x+iy## form.
Hmmm... I'm new to complex number... and, I don't know how to convert ##1-e^{i\pi/3}## to rectangular form since there is a constant (in this case, 1)...
Please help..
 
  • #12
blue_leaf77
Science Advisor
Homework Helper
2,629
784
Hmmm... I'm new to complex number... and, I don't know how to convert ##1-e^{i\pi/3}## to rectangular form since there is a constant (in this case, 1)...
Please help..
In rectangular form of a complex number ##x+iy##, both ##x## and ##y## are real and every complex number can be transformed to this form. To do what you need to do, transform ##e^{i\pi/3}## into rectangular form using Euler formula.
 
  • #13
392
13
In rectangular form of a complex number ##x+iy##, both ##x## and ##y## are real and every complex number can be transformed to this form. To do what you need to do, transform ##e^{i\pi/3}## into rectangular form using Euler formula.
Alright,
so it becomes like this
##1-e^{i\pi/3} = 1 - cos \frac{\pi}{3} + i\ sin \frac{\pi}{3} = \frac{1}{2} + \frac{1}{2}\sqrt{3}i = \frac{1}{2} (1+\sqrt{3}i) \\
sin\ x = \sqrt{\frac{1}{2} (1+\sqrt{3}i)} ##

and I convert cos x to rectangular form,

## cos\ x = \frac{1}{2}\sqrt{3}+\frac{1}{2}i##

##sin\ 2x = 2\ sin\ x\ cos\ x\ = 2 (\sqrt{\frac{1}{2} (1+\sqrt{3}i)} )( \frac{1}{2}\sqrt{3}+\frac{1}{2}i)##

Still seems complicated :(
 
  • #14
blue_leaf77
Science Advisor
Homework Helper
2,629
784
You make a mistake ##e^{i\pi/3} = \frac{1}{2} + i\frac{\sqrt{3}}{2}##. After incorporating this correction, you will get ##\sin^2 x## in rectangular form, then cast it back into polar form. Take its square root to obtain ##\sin x##.
 
  • #15
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,796
1,668

Homework Statement



If ##cos^3x+\frac{1}{cos^3x} = 0##, then sin 2x equals ...

A. -1
B. -√3
C. -√2
D. 1
E. 2

Homework Equations


Basic trigonometry

The Attempt at a Solution


[/B]
Let y = cos^3 x

y+1/y = 0
y^2 + 1 = 0
y = √(-1)

Then, I don't know what to do.. Help please..
After looking at all the complex algebra and trig which has been expended on solving this problem in the posts above, you can eliminate some of the answers A. - E., at least in terms of real solutions for x.

The problem statement wants to know what the value of sin (2x) is, and like all sine functions, its amplitude is bound to the interval [-1, 1], so that means that any choices whose value lies outside this range cannot be correct. The remaining choices can be easily checked to see if they can satisfy the original cosine equation.

Given the comments made so far in this thread, I can't help but wonder if something has been left out of the original problem statement.
 
  • #16
blue_leaf77
Science Advisor
Homework Helper
2,629
784
like all sine functions, its amplitude is bound to the interval [-1, 1],
If the argument is complex, then it can be bigger than unity. For example ##\cos (iy) = \cosh y \geq 1##.
 
  • #17
392
13
You make a mistake ##e^{i\pi/3} = \frac{1}{2} + i\frac{\sqrt{3}}{2}##. After incorporating this correction, you will get ##\sin^2 x## in rectangular form, then cast it back into polar form. Take its square root to obtain ##\sin x##.
Alright,
so it becomes like this
##1-e^{i\pi/3} = 1 - cos \frac{\pi}{3} + i\ sin \frac{\pi}{3} = \frac{1}{2} + \frac{1}{2}\sqrt{3}i = \frac{1}{2} (1+\sqrt{3}i) \\
sin\ x = \sqrt{\frac{1}{2} (1+\sqrt{3}i)} ##

and I convert cos x to rectangular form,

## cos\ x = \frac{1}{2}\sqrt{3}+\frac{1}{2}i##

##sin\ 2x = 2\ sin\ x\ cos\ x\ = 2 (\sqrt{\frac{1}{2} (1+\sqrt{3}i)} )( \frac{1}{2}\sqrt{3}+\frac{1}{2}i)##

Still seems complicated :(
I forgot to put parentheses
##1-e^{i\pi/3} = 1 - (cos \frac{\pi}{3} + i\ sin \frac{\pi}{3}) = \frac{1}{2} - \frac{1}{2}\sqrt{3}i = \frac{1}{2} (1-\sqrt{3}i) \\
sin\ x = \sqrt{\frac{1}{2} (1-\sqrt{3}i)}##
##1-e^{i\pi/3} = 1 - (cos \frac{\pi}{3} + i\ sin \frac{\pi}{3}) = \frac{1}{2} - \frac{1}{2}\sqrt{3}i = \frac{1}{2} (1-\sqrt{3}i) \\
sin\ x = \sqrt{\frac{1}{2} (1-\sqrt{3}i)} \\
sin\ 2x = 2\ sin\ x\ cos\ x\ = 2 (\sqrt{\frac{1}{2} (1-\sqrt{3}i)} )( \frac{1}{2}\sqrt{3}+\frac{1}{2}i)##

Stuck again.
 
  • #18
blue_leaf77
Science Advisor
Homework Helper
2,629
784
Like I said, cast ##\frac{1}{2} (1-\sqrt{3}i)## back to polar form.
 
  • #19
392
13
Like I said, cast ##\frac{1}{2} (1-\sqrt{3}i)## back to polar form.
Polar form?? Do you mean something like r (cos alpha + i sin alpha) ?
##r = \sqrt{{\frac{1}{2}}^2+({\frac{1}{2}}\sqrt{3})^2} = \sqrt{1} = 1 \\
\alpha = tan^{-1} (\frac{-\sqrt{3}}{1}) \\
\alpha = \frac{5}{3}\pi \\
\frac{1}{2}(1-\sqrt{3}i) = 1 (cos \frac{5}{3}\pi+i sin\frac{5}{3}\pi) \\##

Taking the root of it?? Why should I make it in polar form??
 
  • #20
blue_leaf77
Science Advisor
Homework Helper
2,629
784
I mean complex exponential form, ##re^{i\theta}##.
Taking the root of it?? Why should I make it in polar form??
You don't need to actually but doing so will simplify your calculation.
 
  • #21
392
13
I mean complex exponential form, ##re^{i\theta}##.
So,
##\frac{1}{2}(1-\sqrt{3}i) = \frac{1}{2} - \frac{1}{2} \sqrt{3}i = e^{i\frac{5\pi}{3}} \\ ##
##
sin\ x= e^{i\frac{5\pi}{6}} \\
sin\ 2x = 2\ sin\ x\ cos\ x = 2 e^{i\frac{5\pi}{6}} e^{i\frac{\pi}{6}} = 2e^{i\pi} = -2
##
Is it correct ??
 
  • #22
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,796
1,668
If the argument is complex, then it can be bigger than unity. For example ##\cos (iy) = \cosh y \geq 1##.
Yeah, but I still wonder about this problem, especially since the OP claims he's never worked with complex numbers much.
 
  • #23
blue_leaf77
Science Advisor
Homework Helper
2,629
784
-2 is surely one of the possibilities, but it's not among the choices. You can end up with a different solution if you use another angle instead of ##5\pi/3##, hint: consider a negative angle.
 
  • Like
Likes terryds
  • #24
392
13
-2 is surely one of the possibilities, but it's not among the choices. You can end up with a different solution if you use another angle instead of ##5\pi/3##, hint: consider a negative angle.
##sin\ x = e^{\frac{-\pi}{6}i}##
##sin\ 2x = 2 e^{\frac{-\pi}{6}i} e^{\frac{\pi}{6}i} = 2##
The answer is E. 2... Thanks a lot, pal!
 
  • #25
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
35,524
6,429
##sin\ x = e^{\frac{-\pi}{6}i}##
##sin\ 2x = 2 e^{\frac{-\pi}{6}i} e^{\frac{\pi}{6}i} = 2##
The answer is E. 2... Thanks a lot, pal!
Having got to ##\cos (x)=e^{i\frac{\pi}6}##, I think the easiest way is to look at sin2(2x). ##\sin^2(x)\cos^2(x)=\cos^2(x)-\cos^4(x)=e^{i\frac{\pi}3}-e^{2i\frac{\pi}3}=\cos(\pi/3)+i\sin(\pi/3)-\cos(2\pi/3)-i\sin(2\pi/3)=2\cos(\pi/3)=1##.
 
  • Like
Likes terryds

Related Threads on Trigonometry problem

  • Last Post
Replies
3
Views
769
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
10
Views
952
  • Last Post
Replies
22
Views
2K
  • Last Post
Replies
9
Views
837
  • Last Post
Replies
2
Views
940
  • Last Post
Replies
15
Views
1K
  • Last Post
Replies
4
Views
718
  • Last Post
Replies
5
Views
894
  • Last Post
Replies
1
Views
1K
Top