the original problem was:
Solve: sin(3x)-sin(x)=cos(2x)
so far i've gooten to:
sin(x)(2sin(x)cos(x)-1)=cos^2(x)-sin^2(x)
Where would I go from here?
To continue solving the equation, you can simplify the expression on the right side of the equation and factor out sin(x) from the left side.
Starting with the equation:
sin(x) * (2sin(x)cos(x) - 1) = cos^2(x) - sin^2(x)
Next, simplify the right side using the trigonometric identity cos^2(x) - sin^2(x) = cos(2x):
sin(x) * (2sin(x)cos(x) - 1) = cos(2x)
Now, let's factor out sin(x) from the left side of the equation:
sin(x) * (2cos(x)sin(x) - 1) = cos(2x)
Now, you have two factors on the left side: sin(x) and the expression (2cos(x)sin(x) - 1). Since sin(x) cannot be zero (since it is a trigonometric function), you can divide both sides by sin(x):
2cos(x)sin(x) - 1 = cos(2x) / sin(x)
Now, you can simplify the right side by using the trigonometric identity cos(2x) / sin(x) = 2cos(x):
2cos(x)sin(x) - 1 = 2cos(x)
At this point, you have a linear equation in terms of cos(x). Rearrange the equation to isolate cos(x):
2cos(x)sin(x) - 2cos(x) = 1
Factor out cos(x):
cos(x)(2sin(x) - 2) = 1
Divide both sides by (2sin(x) - 2):
cos(x) = 1 / (2sin(x) - 2)
Now, you can solve for cos(x) by substituting the values of sin(x):
-1 ≤ sin(x) ≤ 1
Substitute sin(x) = 1:
cos(x) = 1 / (2 - 2)
cos(x) = 1 / 0
Since division by zero is undefined, this value is not a solution.
Substitute sin(x) = -1:
cos(x) = 1 / (2 - (-2))
cos(x) = 1 / 4
cos(x) = 0.25
Now, you have the value of cos(x). To find the value of x, you can take the inverse cosine (cos^-1) or arccos of both sides:
x = cos^-1(0.25)
Using a calculator, find the angle whose cosine is 0.25. This will give you the value of x.
Note: Depending on the range in which you are solving the equation, there may be multiple solutions for x. Make sure to consider all possible solutions.