Convert to Trigonometric Form (1-cos(2x))/(sin(2x))

$\frac{1 - \cos (2 x)}{\sin (2 x)}$

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Substitute the actual values of $a = \frac{1 - \cos (2 x)}{\sin (2 x)}$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(\frac{1 - \cos (2 x)}{\sin (2 x)})}^{2}}$

Find $| z |$ .

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Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(\frac{1 - \cos (2 x)}{\sin (2 x)})}^{2}}$

Apply the product rule to $\frac{1 - \cos (2 x)}{\sin (2 x)}$ .

$| z | = \sqrt{0 + \frac{{(1 - \cos (2 x))}^{2}}{\sin^{2} (2 x)}}$

Multiply by $1$ .

$| z | = \sqrt{0 + \frac{{(1 - \cos (2 x))}^{2}}{\sin^{2} (2 x) \cdot 1}}$

Separate fractions.

$| z | = \sqrt{0 + \frac{{(1 - \cos (2 x))}^{2}}{1} \cdot \frac{1}{\sin^{2} (2 x)}}$

Convert from $\frac{1}{\sin^{2} (2 x)}$ to $\csc^{2} (2 x)$ .

$| z | = \sqrt{0 + \frac{{(1 - \cos (2 x))}^{2}}{1} \cdot \csc^{2} (2 x)}$

Simplify the expression.

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Divide ${(1 - \cos (2 x))}^{2}$ by $1$ .

$| z | = \sqrt{0 + {(1 - \cos (2 x))}^{2} \csc^{2} (2 x)}$

Rewrite ${(1 - \cos (2 x))}^{2}$ as $(1 - \cos (2 x)) (1 - \cos (2 x))$ .

$| z | = \sqrt{0 + (1 - \cos (2 x)) (1 - \cos (2 x)) \csc^{2} (2 x)}$

Expand $(1 - \cos (2 x)) (1 - \cos (2 x))$ using the FOIL Method.

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Apply the distributive property.

$| z | = \sqrt{0 + (1 (1 - \cos (2 x)) - \cos (2 x) (1 - \cos (2 x))) \csc^{2} (2 x)}$

Apply the distributive property.

$| z | = \sqrt{0 + (1 \cdot 1 + 1 (- \cos (2 x)) - \cos (2 x) (1 - \cos (2 x))) \csc^{2} (2 x)}$

Apply the distributive property.

$| z | = \sqrt{0 + (1 \cdot 1 + 1 (- \cos (2 x)) - \cos (2 x) \cdot 1 - \cos (2 x) (- \cos (2 x))) \csc^{2} (2 x)}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

$| z | = \sqrt{0 + (1 + 1 (- \cos (2 x)) - \cos (2 x) \cdot 1 - \cos (2 x) (- \cos (2 x))) \csc^{2} (2 x)}$

Multiply $- \cos (2 x)$ by $1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) \cdot 1 - \cos (2 x) (- \cos (2 x))) \csc^{2} (2 x)}$

Multiply $- 1$ by $1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) - \cos (2 x) (- \cos (2 x))) \csc^{2} (2 x)}$

Multiply $- \cos (2 x) (- \cos (2 x))$ .

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Multiply $- 1$ by $- 1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) + 1 \cos (2 x) \cos (2 x)) \csc^{2} (2 x)}$

Multiply $\cos (2 x)$ by $1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) + \cos (2 x) \cos (2 x)) \csc^{2} (2 x)}$

Raise $\cos (2 x)$ to the power of $1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) + \cos (2 x) \cos (2 x)) \csc^{2} (2 x)}$

Raise $\cos (2 x)$ to the power of $1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) + \cos (2 x) \cos (2 x)) \csc^{2} (2 x)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) + {\cos (2 x)}^{1 + 1}) \csc^{2} (2 x)}$

Add $1$ and $1$ .

$| z | = \sqrt{0 + (1 - \cos (2 x) - \cos (2 x) + \cos^{2} (2 x)) \csc^{2} (2 x)}$

Subtract $\cos (2 x)$ from $- \cos (2 x)$ .

$| z | = \sqrt{0 + (1 - 2 \cos (2 x) + \cos^{2} (2 x)) \csc^{2} (2 x)}$

Apply the distributive property.

$| z | = \sqrt{0 + 1 \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cos^{2} (2 x) \csc^{2} (2 x)}$

Simplify.

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Multiply $\csc^{2} (2 x)$ by $1$ .

$| z | = \sqrt{0 + \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cos^{2} (2 x) \csc^{2} (2 x)}$

Rewrite $\csc (2 x)$ in terms of sines and cosines.

$| z | = \sqrt{0 + \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cos^{2} (2 x) {(\frac{1}{\sin (2 x)})}^{2}}$

Apply the product rule to $\frac{1}{\sin (2 x)}$ .

$| z | = \sqrt{0 + \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cos^{2} (2 x) (\frac{1^{2}}{\sin^{2} (2 x)})}$

One to any power is one.

$| z | = \sqrt{0 + \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cos^{2} (2 x) (\frac{1}{\sin^{2} (2 x)})}$

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

$| z | = \sqrt{0 + \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \frac{\cos^{2} (2 x)}{\sin^{2} (2 x)}}$

Convert from $\frac{\cos^{2} (2 x)}{\sin^{2} (2 x)}$ to $\cot^{2} (2 x)$ .

$| z | = \sqrt{0 + \csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cot^{2} (2 x)}$

Add $0$ and $\csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cot^{2} (2 x)$ .

$| z | = \sqrt{\csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cot^{2} (2 x)}$

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{0}{\frac{1 - \cos (2 x)}{\sin (2 x)}})$

Substitute the values of $θ = \arctan (\frac{0}{\frac{1 - \cos (2 x)}{\sin (2 x)}})$ and $| z | = \sqrt{\csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cot^{2} (2 x)}$ .

$\sqrt{\csc^{2} (2 x) - 2 \cos (2 x) \csc^{2} (2 x) + \cot^{2} (2 x)} (\cos (\arctan (\frac{0}{\frac{1 - \cos (2 x)}{\sin (2 x)}})) + i \sin (\arctan (\frac{0}{\frac{1 - \cos (2 x)}{\sin (2 x)}})))$

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