Solve sin(x) = cos(2x) for all x

Hello,

The question is: Solve sin(x) = cos(2x) for all x.

My solution:

sin(x) = cos(2x)

sin(x) = sin(1/2 pi - 2x)

x = 1/2 pi - 2x

3x = 1/2 pi

x = 1/6 pi

However, the question says "solve for all x". I understand that fact that there are multiple x-es since both functions are periodic. But how do I find the other x-es? Please help!

Re: Solve sin(x) = cos(2x) for all x

Quote:

Originally Posted by

**Lotte1990** Hello,

The question is: Solve sin(x) = cos(2x) for all x.

My solution:

sin(x) = cos(2x)

sin(x) = sin(1/2 pi - 2x)

x = 1/2 pi - 2x

3x = 1/2 pi

x = 1/6 pi

However, the question says "solve for all x". I understand that fact that there are multiple x-es since both functions are periodic. But how do I find the other x-es? Please help!

It would be better to use the identity , so that your equation becomes

Now solve this quadratic.

Re: Solve sin(x) = cos(2x) for all x

or

or

or

or or in which k is an integer.

Is this correct? And is this the COMPLETE answer? Thanks!

Re: Solve sin(x) = cos(2x) for all x

Yes, that is the complete answer.

Re: Solve sin(x) = cos(2x) for all x

Ok, then please explain to me why I find x equal to 1/6 pi in the first post, but not in the second one. I don't understand...

Re: Solve sin(x) = cos(2x) for all x

solutions for the interval ...

...

...

Re: Solve sin(x) = cos(2x) for all x

Quote:

Originally Posted by

**Lotte1990** Ok, then please explain to me why I find x equal to 1/6 pi in the first post, but not in the second one. I don't understand...

because you assumed that if sin(x) = sin(pi/2 - 2x), that x = pi/2 - 2x. but this is not true, sin(x) is not a 1-1 function.

for example, sin(pi/2 - 2x) = sin(5pi/2 - 2x) as well (leading to x = 5pi/6), and also cos(2x) = cos(-2x), so

we have sin(x) = sin(pi/2 + 2x) also (leading to x = -pi/2, and thus (by adding 2pi) x = 3pi/2).

it is customary to find the solutions "in one revolution" (either from -pi to pi, or 0 to 2pi), and then add 2kpi to the results.

Re: Solve sin(x) = cos(2x) for all x

Quote:

Originally Posted by

**Lotte1990**

You have solved the quadratic equation incorrectly.