1. ## Trig equations: cos4x=sin2x

I have to solve cos4x=sin2x where x is an element of 0, 360.

The problem, for me, is cos4x. I haven't proved this myself, but a quick search gave me that cos4x=$\displaystyle 8cos^4x-8cos^2x+1$

By going along the lines of $\displaystyle 8cos^4x-8cos^2x+1=2sinxcosx$ (see original question) I managed to arrive at sinx=$\displaystyle \frac{1}{3}$

This does not give an exact value and I was wondering if I had done this correctly. Thanks!

2. Originally Posted by RAz
I have to solve cos4x=sin2x where x is an element of 0, 360.

The problem, for me, is cos4x. I haven't proved this myself, but a quick search gave me that cos4x=$\displaystyle 8cos^4x-8cos^2x+1$

By going along the lines of $\displaystyle 8cos^4x-8cos^2x+1=2sinxcosx$ (see original question) I managed to arrive at sinx=$\displaystyle \frac{1}{3}$

This does not give an exact value and I was wondering if I had done this correctly. Thanks!
$\displaystyle \cos{4x} = \sin{2x}$

$\displaystyle \cos^2{2x} - \sin^2{2x} = \sin{2x}$

$\displaystyle 1 - \sin^2{2x} - \sin^2{2x} = \sin{2x}$

$\displaystyle 1 - 2\sin^2{2x} = \sin{2x}$

$\displaystyle 0 = 2\sin^2{2x} + \sin{2x} - 1$.

You now have a quadratic equation in $\displaystyle \sin{2x}$.

So let $\displaystyle \sin{2x} = X$, so that you can see the Quadratic:

$\displaystyle 0 = 2X^2 + X - 1$

$\displaystyle 0 = 2X^2 + 2X - X - 1$

$\displaystyle 0 = 2X(X + 1) - 1(X + 1)$

$\displaystyle 0 = (X + 1)(2X - 1)$.

So $\displaystyle X = -1$ or $\displaystyle X = \frac{1}{2}$.

This means $\displaystyle \sin{2x} = -1$ or $\displaystyle \sin{2x} = \frac{1}{2}$.

Case 1:

$\displaystyle \sin{2x} = -1$

$\displaystyle 2x = 270^{\circ} + 360^{\circ}n$, where $\displaystyle n$ is an integer.

$\displaystyle x = 135^{\circ} + 180^{\circ}n$, where $\displaystyle n$ is an integer.

Case 2:

$\displaystyle \sin{2x} = \frac{1}{2}$

Since sine is positive in the first and second quadrants:

$\displaystyle 2x = \left\{30^{\circ}, 180^{\circ} - 30^{\circ} \right\} + 360^{\circ} n$

$\displaystyle 2x = \left\{ 30^{\circ}, 150^{\circ}\right\} + 360^{\circ} n$

$\displaystyle x = \left \{ 15^{\circ}, 75^{\circ} \right\} + 180^{\circ} n$.

So putting it together:

$\displaystyle x = \left\{ 15^{\circ}, 75^{\circ}, 135^{\circ}\right\} + 180^{\circ}n$.

Upon substituting $\displaystyle n = 0$ and $\displaystyle n = 1$, we find all the possible solutions in the domain $\displaystyle 0^{\circ} \leq x \leq 360^{\circ}$:

$\displaystyle x = \left\{ 15^{\circ}, 75^{\circ}, 135^{\circ}, 195^{\circ}, 255^{\circ}, 315^{\circ} \right\}$.

3. Thanks a lot, I appreciate it. Very clear.

4. Just to give an alternative way

$\displaystyle \cos{4x} = \sin{2x}$

$\displaystyle \cos{4x} = \cos{(90^{\circ}-2x)}$

Case 1:
$\displaystyle 4x = 90^{\circ}-2x+ 360^{\circ} n$

$\displaystyle 6x = 90^{\circ} + 360^{\circ} n$

$\displaystyle x = 15^{\circ} + 60^{\circ} n$

$\displaystyle x = \left\{ 15^{\circ}, 75^{\circ}, 135^{\circ}, 195^{\circ}, 255^{\circ}, 315^{\circ} \right\}$

Case 2:
$\displaystyle 4x = -(90^{\circ}-2x) + 360^{\circ} n$

$\displaystyle 4x = -90^{\circ}+2x + 360^{\circ} n$

$\displaystyle 2x = -90^{\circ} + 360^{\circ} n$

$\displaystyle x = -45^{\circ} + 180^{\circ} n$

$\displaystyle x = \left\{135^{\circ}, 315^{\circ} \right\}$

Therefore:
$\displaystyle x = \left\{ 15^{\circ}, 75^{\circ}, 135^{\circ}, 195^{\circ}, 255^{\circ}, 315^{\circ} \right\}$

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# trigonomatry cos4x/cos2y

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