Trigonometry Examples

$8 i + 6$

Step 1

Reorder $8 i$ and $6$ .

$6 + 8 i$

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 $a = 6$ and $b = 8$ .

$| z | = \sqrt{8^{2} + 6^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

Raise $8$ to the power of $2$ .

$| z | = \sqrt{64 + 6^{2}}$

Step 5.2

Raise $6$ to the power of $2$ .

$| z | = \sqrt{64 + 36}$

Step 5.3

Add $64$ and $36$ .

$| z | = \sqrt{100}$

Step 5.4

Rewrite $100$ as $10^{2}$ .

$| z | = \sqrt{10^{2}}$

Step 5.5

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

$| z | = 10$

Step 6

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

$θ = \arctan (\frac{8}{6})$

Step 7

Since inverse tangent of $\frac{8}{6}$ produces an angle in the first quadrant, the value of the angle is $0.92729521$ .

$θ = 0.92729521$

Step 8

Substitute the values of $θ = 0.92729521$ and $| z | = 10$ .

$10 (\cos (0.92729521) + i \sin (0.92729521))$