Trigonometry Examples

\tan (90 - θ)

$\tan (90 - θ)$

Step 1

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

z = a + b i = | z | (\cos (θ) + i \sin (θ))

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 2

The modulus of a complex number is the distance from the origin on the complex plane.

| z | = \sqrt{a^{2} + b^{2}}

$| z | = \sqrt{a^{2} + b^{2}}$ where

z = a + b i

$z = a + b i$

Step 3

Substitute the actual values of

a = \tan (90 - θ)

$a = \tan (90 - θ)$ and

b = 0

$b = 0$ .

| z | = \sqrt{0^{2} + \tan^{2} (90 - θ)}

$| z | = \sqrt{0^{2} + \tan^{2} (90 - θ)}$

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Raising

0

$0$ to any positive power yields

0

$0$ .

| z | = \sqrt{0 + \tan^{2} (90 - θ)}

$| z | = \sqrt{0 + \tan^{2} (90 - θ)}$

Step 4.2

Add

0

$0$ and

\tan^{2} (90 - θ)

$\tan^{2} (90 - θ)$ .

| z | = \sqrt{\tan^{2} (90 - θ)}

$| z | = \sqrt{\tan^{2} (90 - θ)}$

Step 4.3

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

| z | = \tan (90 - θ)

$| z | = \tan (90 - θ)$

| z | = \tan (90 - θ)

$| z | = \tan (90 - θ)$

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}{\tan (90 - θ)})

$θ = \arctan (\frac{0}{\tan (90 - θ)})$

Step 6

Substitute the values of

θ = \arctan (\frac{0}{\tan (90 - θ)})

$θ = \arctan (\frac{0}{\tan (90 - θ)})$ and

| z | = \tan (90 - θ)

$| z | = \tan (90 - θ)$ .

\tan (90 - θ) (\cos (\arctan (\frac{0}{\tan (90 - θ)})) + i \sin (\arctan (\frac{0}{\tan (90 - θ)})))

$\tan (90 - θ) (\cos (\arctan (\frac{0}{\tan (90 - θ)})) + i \sin (\arctan (\frac{0}{\tan (90 - θ)})))$