Trigonometry Examples

$1 - \cos (x)$

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 $b = - 1 \cos (x)$ .

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

Step 4

Find $| z |$ .

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Step 4.1

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 4.2

Apply the product rule to $- \cos (x)$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

One to any power is one.

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

Step 5

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

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

Step 6

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

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