SOLUTION: Please help me with this question and show all steps if possible thank you! {{{2sin2x-cosx=0}}}

Question 312171: Please help me with this question and show all steps if possible thank you!

Answer by nyc_function(2741) (Show Source): You can put this solution on YOUR website!
2sin^2 x - cosx = 0
One of the Pythagorean Identities is sin^2 x = 1 - cos^2 x.
Replace your sin^2 x with 1 - cos^2 x.
2(1 - cos^2 x) - cosx = 0
We now apply the distributive rule from algebra to remove the parentheses.
2 - 2cos^2 x - cosx = 0
Bring everything to the right side of the trig equation and then equate to zero.
0 = 2cos^2 x + cosx - 2
To avoid confusion, replace cosx with any letter of choice to complete the factoring process. Then, at the end, replace the letter your chose with cosx. More about this later.
I will use the letter u but you can use any letter.
2u^2 + u - 2 = 0
Do you recognize that this is a quadratic equation dressed up as a trig function?
We now factor like any other quadratic equation.
To factor, we need to use the quadratic formula from algebra in this case.
After doing the math on paper, I got two answers for u.
Here they are:
u = [-1 + sqrt{17}]/4, which is about 0.781 in decimal form.
AND
u = [-1 - sqrt{17}]/4, which is rejected because the value of cosine must lie between 0 and 1. Do you understand why the negative decimal number must be rejected?

We are working with the positive decimal number. Keep in mind that cosine is positive in the first and fourth quadrants.
We can now replace u with cosx because we are dealing with trigonometry not algebra.
We are working with cosx = 0.781.
Using the inverse cosine key on the calculator, I found the reference angle to be 38 degrees.
Since we are in quadrants 1 and 4, we now find the values of x in those quadrants.
In Quadrant 1:
90 - 38 = 52 degrees
In Quadrant 4:
360 - 38 = 322 degrees
I hope this helps.

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