Finding the Value of 2sin(2x) - \sqrt{3}

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In summary, Homework Equations: Sin(2x) = 2SinxCosx. Sin(2x) = 2sinxcosx. The Attempt at a Solution: I subtracted root3 and then divided everything by 2 which leaves me with sin (2x) = \sqrt{3}/2. I used the double angle identity to change sin 2x into 2sinxcosx. I'm stuck from there. I forgot how to use double angles ( eek.). I wouldn't use the double angle identity to solve this problem.
  • #1
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Homework Statement



2sin(2x) - [tex]\sqrt{3}[/tex] = 0

Find the value of the variable.

Homework Equations



Sin(2x) = 2SinxCosx

The Attempt at a Solution


I subtracted root3 and then divided everything by 2 which leaves me with

sin (2x) = [tex]\sqrt{3}[/tex]/2

then I used the double angle identity to change sin 2x into 2sinxcosx.

I'm stuck from there. I forgot how to use double angles ( eek.)
 
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  • #2
I wouldn't use the double angle identity to solve this problem. Go back to, sin(2x) = (3)1/2/2 and take the inverse sine.
 
  • #3
How do I do that?
 
  • #4
If asked you to find all x that satisfy, sin(x) = 1/2, what would you do? If you can do that problem, then you can do this one.
 
  • #5
I'm still stuck. I used the inverse sin of the (root3)/2 and I got answers like pi/3, 2pi/3, ect.

I think I need to use the identity but it seems impossible to figure out:

2sinx cosx = (root3)/2
 
  • #6
You don't need the identity. You got multiple answers because the trigonmetic functions are periodic so there are an infinite number of solutions to equations like sin(x) = 1/2. You really want to use the inverse sin in this case.
 
  • #7
What happens to the 2x ?
 
  • #8
You end up solving it for x, just the same as if I told you 2x = 6.
 
  • #9
HOLY oh man, I am real angry right now.

I left my calculator in radian mode and I was using degrees for like the last 40 mins !
 
  • #10
Whats significance does the identity: sin2x = 2sinxcosx have?
 
  • #11
I'm said:
Whats significance does the identity: sin2x = 2sinxcosx have?

It doesn't have significance in this problem. It's useful when you need to manipulate trigonometric functions to put them into an integrable form or when you need to prove other identities.
 
  • #12
let t=2x.

can you solve sin(t) = sqrt(3)/2
 
  • #13
yeah, its pi/6, 5pi/6, and so forth.
Whats the sign that I add to the answer if it's periodical? How exactly do I write the answer for this problem? Do I write all the radians that make the equation true one time around the unit circle then add the (periodical)?
 
  • #14
You usually write your answers with k where k is any integer; for example, solutions for sin(x) = 1 is x = pi/2 + 2k*pi
Your answer will need a little more than just 2k*pi
 
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  • #15
You should end up with multiple answers as said before trig functions are periodical.
Did the question have any limits? i.e 0<x<2pi/360??
 
  • #16
No limits.

So my answer is basicaly, 30 degrees or pi/6 + 2kpi? Correct?
 
  • #17
pi/6 + k*pi takes care of half of the answers; you also need the solutions with 5pi/6.
 
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  • #18
[tex]sin(2x)=\frac{\sqrt{3}}{2}[/tex]

[tex]2x=\begin{bmatrix}
\frac{\pi}{3} + 2k\pi\\
\pi-\frac{\pi}{3}+2k\pi
\end{bmatrix}[/tex]

Can you solve it now?
 
  • #19
x = pi/6 + 2kpi, 5pi/6 = 2kpi?

On a test or something, would I just leave the answers as so? ( if their correct).
 
  • #20
No.
Starting with Дьявол's work (which I haven't checked, but believe is correct), you have
2x = pi/3 + 2k*pi, or 2x = 2pi/3 + 2k*pi, so
x = pi/6 + k*pi, or x = pi/3 + k*pi, where k is any integer.

You forgot to divide the 2k*pi parts.
 
  • #21
Mark44 you're right. The final answers are:
x = pi/6 + k*pi, or x = pi/3 + k*pi, where k is any integer.

He forgot to divede the 2k*pi parts.

Regards.
 
  • #22
Yups
 
  • #23
hullo..??...
sin(2x)=sqrt(3/4)..u can get a genenarl solution 4 dat..the identity is unnecessary over here..
sin t=sin(a)
then t=n(pi)+((-1)^n)a
use this and solve 4 "x"
 
  • #24
I believe that was said about 22 posts earlier!
 
  • #25
"HAPPY REALIZATION"..lolzz.xD
 

1. What is the general formula for finding the value of 2sin(2x) - √3?

The general formula for finding the value of 2sin(2x) - √3 is 2sin(2x) - √3 = 2sin(2x) - 1.732.

2. How can I simplify the expression 2sin(2x) - √3?

To simplify the expression 2sin(2x) - √3, you can use the trigonometric identity sin(2x) = 2sin(x)cos(x). This will result in the simplified expression 4sin(x)cos(x) - √3.

3. Can the value of 2sin(2x) - √3 be negative?

Yes, the value of 2sin(2x) - √3 can be negative. This will depend on the value of x and the quadrant in which it lies. If x is in the second or third quadrant, then the value will be negative. Otherwise, if x is in the first or fourth quadrant, the value will be positive.

4. How do I find the exact value of 2sin(2x) - √3?

To find the exact value of 2sin(2x) - √3, you will need to know the exact values of sin(x) and cos(x) for the given value of x. You can use a calculator or refer to a trigonometric table to find these values. Once you have the exact values of sin(x) and cos(x), you can substitute them into the simplified expression 4sin(x)cos(x) - √3 to find the exact value.

5. How is the value of 2sin(2x) - √3 related to the unit circle?

The value of 2sin(2x) - √3 is related to the unit circle through the trigonometric functions sine and cosine. For a given value of x, the expression 2sin(2x) represents the y-coordinate of a point on the unit circle, while √3 represents the length of the line connecting the origin to that point. The difference between these two values represents the vertical distance between the point and the x-axis, which can be thought of as the vertical component of the vector representing the point's position on the unit circle.

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