Trigonometry Examples

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

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 $a = 1$ and $b = - \frac{1 \cos^{2} (x)}{1 + \sin (x)}$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Raise $- 1$ to the power of $2$ .

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

Step 4.4

Multiply $\frac{{(\cos^{2} (x))}^{2}}{{(1 + \sin (x))}^{2}}$ by $1$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Multiply $2$ by $2$ .

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

Step 4.6

One to any power is one.

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

Step 5

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

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

Step 6

Substitute the values of $θ = \arctan (\frac{- \frac{1 \cos^{2} (x)}{1 + \sin (x)}}{1})$ and $| z | = \sqrt{\frac{\cos^{4} (x)}{{(1 + \sin (x))}^{2}} + 1}$ .

$\sqrt{\frac{\cos^{4} (x)}{{(1 + \sin (x))}^{2}} + 1} (\cos (\arctan (\frac{- \frac{1 \cos^{2} (x)}{1 + \sin (x)}}{1})) + i \sin (\arctan (\frac{- \frac{1 \cos^{2} (x)}{1 + \sin (x)}}{1})))$