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 =

Algebra ->  Trigonometry-basics -> 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 =       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
Let y=sin%28x%29

So the equation becomes

2y%5E2+-+6y+-+3+=+0

Now use the quadratic formula to solve for y
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ay%5E2%2Bby%2Bc=0 (in our case 2y%5E2%2B-6y%2B-3+=+0) has the following solutons:

y%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=%28-6%29%5E2-4%2A2%2A-3=60.

Discriminant d=60 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+60+%29%29%2F2%5Ca.

y%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+60+%29%29%2F2%5C2+=+3.43649167310371
y%5B2%5D+=+%28-%28-6%29-sqrt%28+60+%29%29%2F2%5C2+=+-0.436491673103709

Quadratic expression 2y%5E2%2B-6y%2B-3 can be factored:
2y%5E2%2B-6y%2B-3+=+2%28y-3.43649167310371%29%2A%28y--0.436491673103709%29
Again, the answer is: 3.43649167310371, -0.436491673103709. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-6%2Ax%2B-3+%29



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

y=-0.436491673103709 or y=3.43649167310371

which means

sin%28x%29=-0.436491673103709 or sin%28x%29=3.43649167310371

For each case, take the arcsine of both sides
arcsin%28sin%28x%29%29=arcsin%28-0.436491673103709%29 or arcsin%28sin%28x%29%29=arcsin%283.43649167310371%29

taking the arcsine of the first answer, we get
x=arcsin%28-0.436491673103709%29=-0.45169556320258%2B2pi%2An or

x=arcsin%28-0.436491673103709%29=pi--0.45169556320258=3.59328821679238%0D%0A 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
x=arcsin%283.43649167310371%29=undefined



So our answer is
x=-0.45169556320258%2B2pi%2An or x=3.59328821679238%2B2pi%2An