How can we find all the solutions to the equation sin²(2x) sin(x) = 0?

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To solve the equation sin²(2x) sin(x) = 0, we need to consider when either factor of the equation equals zero. The equation consists of two parts, sin²(2x) and sin(x). Let’s break it down.

Step 1: Set each factor to zero

The equation is satisfied if either of the following holds true:

  • sin²(2x) = 0
  • sin(x) = 0

Step 2: Solve sin(x) = 0

The sine function is equal to zero at integer multiples of π:

x = nπ, where n is an integer.

Step 3: Solve sin²(2x) = 0

Since sin²(2x) = 0, we can set sin(2x) = 0:

The solutions for sine equaling zero are:

2x = mπ, where m is an integer.

Dividing both sides by 2 gives us:

x = (m/2)π.

Step 4: Combine both sets of solutions

Now that we have found the solutions from both factors, we can combine them:

  • From sin(x) = 0: x = nπ (where n is an integer)
  • From sin(2x) = 0: x = (m/2)π (where m is an integer)

Final Solution

The complete set of solutions is:

x = nπ

and

x = (m/2)π

Where n and m are any integers. This means that the solutions include both integer multiples of π and half-integer multiples, expanding our solution set considerably.

In summary, to find all solutions to the equation sin²(2x) sin(x) = 0, we consider both factors equating to zero, leading to a comprehensive solution set.

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