Solve for k sin(4x)^2=1/2(1-cos(kx))

$\sin^{2} (4 x) = \frac{1}{2} (1 - \cos (k x))$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (1 - \cos (k x)) = \sin^{2} (4 x)$ .

$\frac{1}{2} \cdot (1 - \cos (k x)) = \sin^{2} (4 x)$

Step 2

Subtract $\sin^{2} (4 x)$ from both sides of the equation.

$\frac{1}{2} \cdot (1 - \cos (k x)) - \sin^{2} (4 x) = 0$

Step 3

Simplify each term.

Apply the distributive property.

$\frac{1}{2} \cdot 1 + \frac{1}{2} (- \cos (k x)) - \sin^{2} (4 x) = 0$

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} + \frac{1}{2} (- \cos (k x)) - \sin^{2} (4 x) = 0$

Combine $\frac{1}{2}$ and $\cos (k x)$ .

$\frac{1}{2} - \frac{\cos (k x)}{2} - \sin^{2} (4 x) = 0$

Step 4

Replace $\sin^{2} (4 x)$ with $1 - \cos^{2} (4 x)$ .

$\frac{1}{2} - \frac{\cos (k x)}{2} - (1 - \cos^{2} (4 x)) = 0$

Step 5

Simplify the left side.

Apply pythagorean identity.

$\frac{1}{2} - \frac{\cos (k x)}{2} - \sin^{2} (4 x) = 0$

Step 6

Replace $\sin^{2} (4 x)$ with $1 - \cos^{2} (4 x)$ .

$\frac{1}{2} - \frac{\cos (k x)}{2} - (1 - \cos^{2} (4 x)) = 0$

Step 7

Apply pythagorean identity.

$\frac{1}{2} - \frac{\cos (k x)}{2} - \sin^{2} (4 x) = 0$

Step 8

Replace $\sin^{2} (4 x)$ with $1 - \cos^{2} (4 x)$ .

$\frac{1}{2} - \frac{\cos (k x)}{2} - (1 - \cos^{2} (4 x)) = 0$

Step 9

Move all terms not containing $k$ to the right side of the equation.

Subtract $\frac{1}{2}$ from both sides of the equation.

$- \frac{\cos (k x)}{2} - (1 - \cos^{2} (4 x)) = - \frac{1}{2}$

Add $1 - \cos^{2} (4 x)$ to both sides of the equation.

$- \frac{\cos (k x)}{2} = - \frac{1}{2} + 1 - \cos^{2} (4 x)$

Apply pythagorean identity.

$- \frac{\cos (k x)}{2} = - \frac{1}{2} + \sin^{2} (4 x)$

Step 10

Move all the expressions to the left side of the equation.

Add $\frac{1}{2}$ to both sides of the equation.

$- \frac{\cos (k x)}{2} + \frac{1}{2} = \sin^{2} (4 x)$

Subtract $\sin^{2} (4 x)$ from both sides of the equation.

$- \frac{\cos (k x)}{2} + \frac{1}{2} - \sin^{2} (4 x) = 0$

Step 11

Replace $\sin^{2} (4 x)$ with $1 - \cos^{2} (4 x)$ .

$- \frac{\cos (k x)}{2} + \frac{1}{2} - (1 - \cos^{2} (4 x)) = 0$

Solve for k sin(4x)^2=1/2(1-cos(kx))

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