SOLUTION: Cos 2x + cos x + 1= 0. Solve.

Question 558191: Cos 2x + cos x + 1= 0. Solve.
Found 2 solutions by Alan3354, richard1234:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Cos 2x + cos x + 1= 0. Solve.
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Sub u for cos)x)

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

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 : .

The discriminant -3 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

In the field of imaginary numbers, the square root of -3 is + or - .

The solution is , or
Here's your graph:

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Sub back:
cos(x) = -1/2 +/- i*sqrt(3)/2
No real solutions.

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
I presume you mean instead of .

Using trig identities, we have

Regroup this way:

Now sin^2 - 1 is equal to -cos^2 x, so we have

If cos x = 0 then x = pi/2, 3pi/2, 5pi/2, etc. Otherwise, we may divide by cos x to obtain

We obtain x = 2pi/3, 4pi/3, 8pi/3, 10pi/3, ... or in general

or or where k is any integer.
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