SOLUTION: how to solve for x if 2sin^2x-3sinx=1

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Question 261029: how to solve for x if 2sin^2x-3sinx=1
Found 2 solutions by Greenfinch, jsmallt9:
Answer by Greenfinch(383) About Me  (Show Source):
You can put this solution on YOUR website!
The expression you give does not gives answers in the ranges -1 to +
If you change it to the following (by changing the 1 to -)
2(sin^2 x) - 3 sin x + 1 = 0
This factorizes to (2sin x - 1 )(sin x - 1 ) = 0
so sinx = 0.5 or 1
x = 30, 90, 150 degrees in the range 0 < x < 360
If you have a -1 instead of +1, the results are (3+-(sqrt 17))/2 which is very messy

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since the equation you posted has a terribly messy solution, I am going to assume that the equation is actually 2sin%5E2%28x%29-3sin%28x%29+=+-1

Would you know how to solve 2q%5E2+-+3q+=+-1? I ask because solving your equation involves
  • Knowing how to solve my equation; and
  • Recognizing that your equation has the same structure as my equation.

First let's look at my equation. It is a quadratic equation. So we solve it by getting one side equal to zero:
2q%5E2+-+3q+%2B+1+=+0
and then factoring the left side (or using the Quadratic Formula). This factors fairly easily:
%282q+-+1%29%28q+-+1%29+=+0
By the Zero Product Property, this product is zero only if one of the factors is zero:
2q+-+1+=+0 or q+-+1+=+0
Solving these we get:
q+=+1%2F2 or q+=+1

Next you need to see that your equation and mine have the same structure. They both say: "2 times something squared minus 3 times that something equals -1". If you still have trouble seeing this, then use a temporary variable:
Let q = sin(x)
Then substitute this variable into your equation for sin(x):
2%28q%29%5E2+-+3%2A%28q%29+=+-1
which is my equation! So you will end up with my solution:
q+=+1%2F2 or q+=+1
Of course your are trying to solve for x, not the temporary variable q. So substitute sin(x) back in for q:
sin%28x%29+=+1%2F2 or sin%28x%29+=+1

We still haven't solved for x but we're getting closer. Everything we have done so far is Algebra. At this point we finally need some Trig. We have to know when the sin function is 1/2 and when it is 1. If we know the special angles we know that for sin(x) = 1/2:
x+=+pi%2F6+%2B+2pi%2An or x+=+5pi%2F6+%2B+2pi%2An}}}
and for sin(x) = 1:
x+=+pi%2F2+%2B+2pi%2An

(Eventually you will learn how to solve problems like this without use of a temporary variable like q. But until then, feel free to use them knowing that at some point you will need to substitute back for it.)