SOLUTION: 3 sin 2θ − 2 sin θ = 0 Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where

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Question 595779: 3 sin 2θ − 2 sin θ = 0
Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
Answer by jsmallt9(3759)   (Show Source): You can put this solution on YOUR website!
(Note: To save typing I'm going to use "x" instead of theta. Just replace my x's with theta's. Also, I'm assuming that the problem is using the convention of theta indicating that the angles are expressed in degrees. If this is not true, then multiply any answers below by to convert them to radian measure.)

3*sin(2x) - 2*sin(x) = 0
Solving equations like this often starts by using algebra and/or trig identities to transform it into one or more equations of the form:
TrigFunction(something) = some-number
For example: sin(x) = 0, tan(4x) = 1, etc.

One of the techniques that is used is to
  1. Get one side of the equation equal to zero
  2. Factor the other side
  3. Use the Zero Product Property
This is the technique we will use on this equation.

We already have a zero on one side. However the other side does not factor as it is written. Another technique that is often used is to use some argument-changing trig identities (2x, (1/2)x, A+B, A-B) to match arguments. We will use the identity: sin(2x) - 2*sin(x)*cos(x). Replacing the sin(2x) in our equation we get:
3*(2*sin(x)cos(x)) - 2*sin(x) = 0
(Note the use of parentheses. This is an extremely good habit when replacing one expression with another!) Simplifying we get:
6*sin(x)cos(x) - 2*sin(x) = 0
All the arguments are now just x, no 2x's are left. And we have an expression that will factor! Factoring out the GCF of 2*sin(x) we get:
2*sin(x)(3*cos(x) - 1) = 0

We can now use the Zero Product Property which tells us that one of these factors must be zero:
2*sin(x) = 0 or 3*cos(x) - 1 = 0
Solving each of these we get:
sin(x) = 0 or cos(x) = 1/3
We now have two equations of the desired form. From these we can find their solutions and, therefore, the solutions to your original equation.

Angles whose sin is zero are special angles and they should be well-known to you. from sin(x) = 0 we should know that:
x = 0 + 360k or x = 180 + 360k

Angles whose cos is 1/3 are not special angles. So we will need our calculators to find the reference angle. Making sure the calculator is set to degree mode and entering should give us a reference angle, rounded to 3 places, of 70.529. Since cos is positive in the 1st and 4th quadrants we get:
x = 70.529 + 360k or x = -70.259 + 360k

This makes the general solution to your equation:
x = 0 + 360k or x = 180 + 360k or x = 70.529 + 360k or x = -70.259 + 360k

P.S.
  1. Problems like this often ask for solutions within a certain range of numbers. This problem did not. But when this is done, always start by finding the general solution (i.e. all possible solutions) and then use various integers for "k" in the general solution equation(s) to find all the specific solutions with the desired range.
  2. The equations x = 0 + 360k and x = 180 + 360k could be "merged" into x = 0 + 180k. See if you tell how this one equation would give you all the same values for x as the original two would give you. (Note: You will not always be able to "merge" equations like this. The other pair of equations will not merge.)
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