Trigonometry Examples

$\cos (x + y)$

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 $a = \cos (x + y)$ and $b = 0$ .

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

Step 4

Find $| z |$ .

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Step 4.1

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

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

Step 4.2

Add $0$ and $\cos^{2} (x + y)$ .

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

Step 4.3

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

$| z | = \cos (x + y)$

Step 5

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

$θ = \arctan (\frac{0}{\cos (x + y)})$

Step 6

Substitute the values of $θ = \arctan (\frac{0}{\cos (x + y)})$ and $| z | = \cos (x + y)$ .

$\cos (x + y) (\cos (\arctan (\frac{0}{\cos (x + y)})) + i \sin (\arctan (\frac{0}{\cos (x + y)})))$