Sin3x = sinx Solve for x.

  • Thread starter smallbadwolf
  • Start date
In summary, the conversation discusses the solution to the equation sin3x - sinx = 0 and how to graph it on a calculator. The solution involves simplifying the equation using trigonometric identities and using the zero product law to find the solutions. The graph may show 7 solutions, but only 3 of them are within the given domain.
  • #1
smallbadwolf
14
0
okay this problem stumped me for a while but here is my work for it, and I just got stuck at the end so if any help can be provided thanks in advance.

sin3x = sinx

sin(2x + x) = sinx

sin2x cos x + cos2x sinx = sinx

2sinx cosx cosx + (2cos^2(x) -1) sinx = sinx

2sinx cos^2(x) + 2cos^2(x) sinx - sinx = sinx

2sinx cos^2(x) + 2cos^2(x) sinx - 2sinx = 0

2sinx (cos^2(x) + cos^2(x) - 2) = 0

2sinx (2cos^2(x) - 2) = 0

4sinx(cos^2(x) -1) = 0

4sinx(-sin^2(x)) = 0

4sinx = 0 , -sin^2x = 0

x = 0 , x = 0, pi, 2pi

Why is it when I graph the equation sin3x - sinx = 0 on my calculator, it comes with 7 solutions when i only have 3?
 
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  • #2
solution

sin(2x+x)-sinx=0

sin2xcosx + cos2xsinx-sinx = 0

2sinxcos^2 x + (cos^2 x - sin^2 x) sinx - sinx = 0

2sinxcos^2 x + cos^2xsinx - sin^3 x -sinx=0

sinx(2cos^2 x + cos^2 x - sin^2 x - 1) = 0

sinx(3cos^2x - sin^2x -1) =0

so u kno sinx= 0, thus 0, 180, and 360 degrees are three of the answers
then for inside parenthesis

3cos^2x - sin^2x - 1 = 0

3cos^2x - (1- cos^2x) - 1 = 0

3cos^2x - 1 + cos^2x - 1 = 0

4cos^2x - 2 = 0

cos^2x = 1/2

so u kno Cos^-1( plus/minus sq root of 2 / 2) = 45, 315, 35, 225 degrees

:zzz: loong problems, well sorta
 
  • #3
2sinx (cos^2(x) + cos^2(x) - 2) = 0

should be

2sinx (cos^2(x) + cos^2(x) - 1) = 0
 
  • #4
Oh okay thanks guys
 
  • #5
smallbadwolf said:
sin3x = sinx

sin(2x + x) = sinx

sin2x cos x + cos2x sinx = sinx...

There is a much simpler method to solve problems like that.

If sin(x) = sin(y) then either

y=x=2k*pi

or y=(pi-x)+2k*pi,

where k is integer (zero included).

y=3x now, so either

3x=x+2k*pi --> x = k*pi

or

3x=(2k+1)*pi -x -->x=(2k+1)*pi/4

ehild
 
  • #6
Maybe you already know this, but just in case, sin[m * pi] = 0 for all m. So, x=m*pi for all m satisfies the (trivial) equation:
sin[m*pi] = sin[3*m*pi] = 0

--
edit: where m is an integer.
 
  • #7
smallbadwolf said:
okay this problem stumped me for a while but here is my work for it, and I just got stuck at the end so if any help can be provided thanks in advance.

sin3x = sinx

sin(2x + x) = sinx

sin2x cos x + cos2x sinx = sinx

2sinx cosx cosx + (2cos^2(x) -1) sinx = sinx

2sinx cos^2(x) + 2cos^2(x) sinx - sinx = sinx

2sinx cos^2(x) + 2cos^2(x) sinx - 2sinx = 0

2sinx (cos^2(x) + cos^2(x) - 2) = 0

2sinx (2cos^2(x) - 2) = 0

4sinx(cos^2(x) -1) = 0

4sinx(-sin^2(x)) = 0

4sinx = 0 , -sin^2x = 0

x = 0 , x = 0, pi, 2pi

Why is it when I graph the equation sin3x - sinx = 0 on my calculator, it comes with 7 solutions when i only have 3?

The problem is that the use of your double angle was incorrect but the idea of solving the proble is correct the thing to do here is
sin3x=sinx
implies sin(2x + x)=sinx

implies sin2xcosx +cos2xsinx - sinx=0

implies 2sinx.cosx.cosx +cos2xsinx - sinx=0

implies sinx(2cos^2(x)+ cos2x - 1)=0

implies sinx(2cos^2(x) + 2cos^2(x) - 1 -1)=0

implies sinx(4cos^2(x) - 2)=0
now using the zero product law

implies sinx=0 or cos^2(x)=1/2

then the equation will solve to be x=0 + n360 or x=+or- 45 + n360 where n lies in Z or integers from there you will sub in integers the will give you solutions that lie in your domain you draw your graph
 

1. How do I solve for x when given the equation Sin3x = sinx?

To solve for x, we can use the trigonometric identity Sin3x = 3Sinx - 4Sin^3x. We can then rearrange the equation to get 4Sin^3x - 2Sinx = 0. From here, we can factor out a Sinx and solve for the remaining quadratic equation to find the possible values of x.

2. What is the domain and range of the given equation Sin3x = sinx?

The domain of the equation is all real numbers, while the range is the interval [-1, 1]. This is because the sine function can take any input value, but outputs values between -1 and 1.

3. Can this equation have multiple solutions for x?

Yes, this equation can have multiple solutions for x. Since the sine function is periodic, it repeats itself every 2π radians or 360 degrees. Therefore, there can be infinite values of x that satisfy the equation.

4. How do I check my solutions for accuracy?

To check your solutions for accuracy, you can plug in the values of x into the original equation and see if both sides are equal. This is a common technique in solving equations and can help catch any potential errors.

5. Can this equation be solved using a calculator?

Yes, you can use a calculator to solve this equation. Most scientific calculators have a sine function and can solve for x using the methods mentioned in the first question. However, it is always good to have a basic understanding of the underlying concepts and identities used in solving the equation.

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