Trigonometry Examples

$\cot (θ) + \tan (θ)$

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 $a = \cot (θ)$ and $b = \tan (θ)$ .

$| z | = \sqrt{\tan^{2} (θ) + \cot^{2} (θ)}$

Step 4

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

$θ = \arctan (\frac{\tan (θ)}{\cot (θ)})$

Step 5

Substitute the values of $θ = \arctan (\frac{\tan (θ)}{\cot (θ)})$ and $| z | = \sqrt{\tan^{2} (θ) + \cot^{2} (θ)}$ .

$\sqrt{\tan^{2} (θ) + \cot^{2} (θ)} (\cos (\arctan (\frac{\tan (θ)}{\cot (θ)})) + i \sin (\arctan (\frac{\tan (θ)}{\cot (θ)})))$