Trigonometry Examples

$i - 5$

Step 1

Reorder $i$ and $- 5$ .

$- 5 + i$

Step 2

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 3

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 4

Substitute the actual values of $a = - 5$ and $b = 1$ .

$| z | = \sqrt{1^{2} + {(- 5)}^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

One to any power is one.

$| z | = \sqrt{1 + {(- 5)}^{2}}$

Step 5.2

Raise $- 5$ to the power of $2$ .

$| z | = \sqrt{1 + 25}$

Step 5.3

Add $1$ and $25$ .

$| z | = \sqrt{26}$

Step 6

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

$θ = \arctan (\frac{1}{- 5})$

Step 7

Since inverse tangent of $\frac{1}{- 5}$ produces an angle in the second quadrant, the value of the angle is $2.94419709$ .

$θ = 2.94419709$

Step 8

Substitute the values of $θ = 2.94419709$ and $| z | = \sqrt{26}$ .

$\sqrt{26} (\cos (2.94419709) + i \sin (2.94419709))$

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