SOLUTION: Solve the trigonometric equation, using exact values when possible, where x is a real number such that 0 is less than or equal to x and less than 2pi. 2sin^2(x) - 6sin(x)

SOLUTION: Solve the trigonometric equation, using exact values when possible, where x is a real number such that 0 is less than or equal to x and less than 2pi. 2sin^2(x) - 6sin(x) - 3 =

Question 84904: Solve the trigonometric equation, using exact values when possible, where x is a real number such that 0 is less than or equal to x and less than 2pi.
2sin^2(x) - 6sin(x) - 3 = 0
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Let

So the equation becomes

Now use the quadratic formula to solve for y

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=60 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 3.43649167310371, -0.436491673103709. Here's your graph:

Since these values are irrational, we cannot use exact values. So we have 2 intermediate answers

or

which means

or

For each case, take the arcsine of both sides
or

taking the arcsine of the first answer, we get
or

remember there are 2 answers when taking the arcsine
Since the domain of arcsine is

we cannot take the arcsine of 3.436. So taking the arcsine of the second answer, we get

So our answer is
or

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