Question

find all solutions to cos(2x)=cosx

Answer 1

Replace cos(2x) with 2cos^2x-1

Answer 2

Hint: Use the double-angle formula for cos(2x) = 2 cos^2 (x) - 1

Answer 3

Set all equal to zero, and factor that quadratic....

Answer 4

Wow!! Wish I would have thought of that.

Answer 5

Thaats why you come to this room!

Answer 6

Factor it like you would factor a quadratic, even though it's not a quadratic.

Answer 7

It IS a quadratic!

Answer 8

2 cos^2(x) - 1 = cos x
2 cos^2(x) - cos x - 1 = 0
(2 cos x + 1)(cos x -1) = 0
cos x = - 1/2 and cos x = 1

Answer 9

I assume you wanted to solve for cos x, and not angles for x, as you haven't given interval for angles, etc.

Answer 10

Sorry, a quadratic is a second degree expression of in one variable. In this case cos x is not a variable, but a function.

Again, it can be factored like a quadratic, but it is not one.

Answer 11

It is a quadratic equation.

Answer 12

Makes no difference if it is x, y, cos x, or whatever you want to call it.

Answer 13

Yes, it does make a difference. A quadratic is a second-degree POLYNOMIAL expression.

In teaching precalculus, I made your mistake for years until corrected by a visiting professor.

2 cos^2(x) - cos x - 1 is a "quadratic type" expression, but not a quadratic.

Is it a fine point? Yes. Is mathematics chock-full of fine points? Yes, again.