SOLUTION: 3sinx-3cosx=1

Algebra ->  Trigonometry-basics -> SOLUTION: 3sinx-3cosx=1      Log On


   

Question 930586: 3sinx-3cosx=1
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
3sinx-3cosx=1
sin - cos = 1/3
sin - sqrt(1 - sin^2) = 1/3
- sqrt(1 - sin^2) = 1/3 - sin
1 - sin^2 = 1/9 - 2sin/3 + sin^2
2sin^2 - 2sin/3 - 8/9 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-0.666666666666667x%2B-0.888888888888889+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-0.666666666666667%29%5E2-4%2A2%2A-0.888888888888889=7.55555555555556.

Discriminant d=7.55555555555556 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--0.666666666666667%2B-sqrt%28+7.55555555555556+%29%29%2F2%5Ca.




Quadratic expression 2x%5E2%2B-0.666666666666667x%2B-0.888888888888889 can be factored:

Again, the answer is: 0.853850937602944, -0.52051760426961. Here's your graph:

=======================
sin(x) =~ 0.853851
x = 1.02334 radians (the principal value)
sin(x) =~ -0.52052
x = 5.7357 radians