Trigonometry Examples

$\sin (θ) + \cos (θ)$

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 $a = \sin (θ)$ and $b = \cos (θ)$ .

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

Step 4

Find $| z |$ .

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Step 4.1

Rearrange terms.

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

Step 4.2

Apply pythagorean identity.

$| z | = \sqrt{1}$

Step 4.3

Any root of $1$ is $1$ .

$| z | = 1$

Step 5

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

$θ = \arctan (\frac{\cos (θ)}{\sin (θ)})$

Step 6

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

$\cos (\arctan (\frac{\cos (θ)}{\sin (θ)})) + i \sin (\arctan (\frac{\cos (θ)}{\sin (θ)}))$