Trigonometry Examples

$\frac{\cos^{2} (θ)}{1 - \sin (θ)}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply the exponents in ${(\cos^{2} (θ))}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.3.2

Multiply $2$ by $2$ .

$| z | = \sqrt{0 + \frac{\cos^{4} (θ)}{{(1 - \sin (θ))}^{2}}}$

Step 4.4

Add $0$ and $\frac{\cos^{4} (θ)}{{(1 - \sin (θ))}^{2}}$ .

$| z | = \sqrt{\frac{\cos^{4} (θ)}{{(1 - \sin (θ))}^{2}}}$

Step 4.5

Rewrite $\cos^{4} (θ)$ as ${(\cos^{2} (θ))}^{2}$ .

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

Step 4.6

Rewrite $\frac{{(\cos^{2} (θ))}^{2}}{{(1 - \sin (θ))}^{2}}$ as ${(\frac{\cos^{2} (θ)}{1 - \sin (θ)})}^{2}$ .

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

Step 4.7

Pull terms out from under the radical, assuming positive real numbers.

$| z | = \frac{\cos^{2} (θ)}{1 - \sin (θ)}$

Step 5

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

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

Step 6

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

$\frac{\cos^{2} (θ)}{(1 - \sin (θ)) (\cos (\arctan (\frac{0}{\frac{\cos^{2} (θ)}{1 - \sin (θ)}})) + i \sin (\arctan (\frac{0}{\frac{\cos^{2} (θ)}{1 - \sin (θ)}})))}$