# Sin3x - sinx = 0 (for x greater than 0 but less than 360)

• Rush147
In summary, the conversation was about solving the equation sin3x - sinx = 0 for x greater than 0 but less than 360, using trigonometrical identities. The suggested methods were using Triple-Angle Formulae, using the properties of sin function, and using the Sum-To-Product Identities. The speaker expressed confusion and asked for clarification and help.
Rush147
Hi folks, can anybody help me. I would like to solve the following equation: ""sin3x - sinx = 0 (for x greater than 0 but less than 360) and come up with angles for x or 2x. We are doing Trigonometrical identities which i have only just come across. Got as far as "sin(2x + x) - sinx = 0" as we think we need to get a 2x or x and sin(2x + x) is also an identity that can be changed, but not sure if this is the right direction.

Many thanks folks

Rush147 said:
"sin(2x + x) - sinx = 0
Can you write the x (in the second term) in terms of 2x and x like you did for 3x?

I believe so as this gives "sin(a + b)" (or in my case sin(2x + x)) and this is equal (or can then be substituted with the identity) "sinacosb + cosasinb", BUT i really don't know if I'm heading the correct way as I've only just this week come across trigonometrical identities so it could be totally different. I know the last equation i did gave me a double angle (2x) which gave 2 angles within 360 degrees. Any help you can give would be most appreciated.

I meant this: sin(2x+x) - sin(2x-x) = 0.

Rush147 said:
I believe so as this gives "sin(a + b)" (or in my case sin(2x + x)) and this is equal (or can then be substituted with the identity) "sinacosb + cosasinb"

Well if you want to try it this way let a=2x and b=x, then use that same formula again on the terms that have 2x's in them. You only want sines from then on...
But it's a lot of algebra

I don't think so as sin(2x+x) could be a backwards step from sin3x but with the "- sinx" there is only one x there so not sure where the 2x and x would come from.

Rush147 said:
I believe so as this gives "sin(a + b)" (or in my case sin(2x + x)) and this is equal (or can then be substituted with the identity) "sinacosb + cosasinb", BUT i really don't know if I'm heading the correct way as I've only just this week come across trigonometrical identities so it could be totally different. I know the last equation i did gave me a double angle (2x) which gave 2 angles within 360 degrees. Any help you can give would be most appreciated.

There are actually 3 ways to solve the above problem.

1. The first way, the most straightforward, and require the most calculation is to trace 3x down to x, by using Triple-Angle Formulae: sin(3x) = 3 sin(x) - 4sin3(x) (You can arrive to this formula by using the Sum-Angle Identity twice). So, your equation becomes a cubic equation in sin(x), which is pretty easy to solve. :)

2. The second way, the easiest way, is to use the properties of sin function:

$$\sin \alpha = \sin \beta$$

$$\Leftrightarrow \left[ \begin{array}{lcr} \alpha & = & \beta + k 360 ^ o \\ \alpha & = & 180 ^ o - \beta + k' 360 ^ o \end{array} \right.$$, k, and k' are both integers.

One can isolate sin(x) to the other side of the equation: sin(3x) = sin(x), and use method mentioned above. Then choose, k, and k' wisely so that your solution is on the interval [0, 360]

3. The 3rd way, the final one, is to use the Sum-To-Product Identities:
(You can arrive to this Itentity by using neutrino's hint)
$$\sin \alpha - \sin \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$$

Hopefully, you can go from here. :)

You can pick up one of the 3 ways mentioned above, or try all 3, and compare the result. :)

Last edited:
I'm not sure if that's at all the correct way but a friend of mine started by using sin(a + b) as this was an identity that could be replaced. These trig identities are all new to me and not the best at algebra either

Thanks for your help. Its all very confusing as I'm quite new to this level of maths. I will have a look through.

## 1. How do you solve the equation Sin3x - sinx = 0 for x greater than 0 but less than 360?

To solve this equation, you can use the trigonometric identity sin(A-B) = sinAcosB - cosAsinB. Substituting A = 3x and B = x, we get: sin(3x-x) = sin3xcosx - cos3xsinx. Expanding the left side, we get: sin2x = sin3xcosx - cos3xsinx. Now, using the double angle identities cos2x = cos^2x - sin^2x and sin2x = 2sinxcosx, we get: 2sinxcosx = sin3xcosx - (1-2cos^2x)sinx. Simplifying further, we get: sinxcosx = sin3xcosx - sinx + 2cos^2xsinx. Factoring out sinx, we get: sinx(cosx + 1) = sin3xcosx + 2cos^2xsinx. Dividing both sides by cosx, we get the final solution: sinx = sin3x + 2cosxsinx. Therefore, the solutions for x greater than 0 but less than 360 are x = 0, 60, 180, 240, and 360 degrees.

## 2. What is the significance of x being greater than 0 but less than 360 in this equation?

The range of 0 to 360 degrees is commonly used for angles in trigonometry. This range represents one full revolution or cycle on a unit circle. Therefore, the solutions for this equation within this range will give us all the possible solutions for one full revolution of the angle.

## 3. Can this equation be solved using any other methods?

Yes, there are other methods for solving this equation, such as using a graphing calculator or using the inverse sine function. However, the trigonometric identity method is the most commonly used and reliable method for solving such equations.

## 4. What does the solution to this equation represent?

The solutions to this equation represent the values of x that satisfy the equation. In other words, when you substitute these values for x in the equation, it will hold true and the left side will equal the right side.

## 5. Is this equation used in any real-world applications?

Yes, trigonometric equations like this one are used in various fields such as physics, engineering, and astronomy. For example, this equation can be used to calculate the position of an object moving in a circular motion or to analyze the behavior of waves in physics.

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