Trigonometry Examples

$1 + i$

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 $a = 1$ and $b = 1$ .

$| z | = \sqrt{1^{2} + 1^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

One to any power is one.

$| z | = \sqrt{1 + 1^{2}}$

Step 4.2

One to any power is one.

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

Step 4.3

Add $1$ and $1$ .

$| z | = \sqrt{2}$

Step 5

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

$θ = \arctan (\frac{1}{1})$

Step 6

Since inverse tangent of $\frac{1}{1}$ produces an angle in the first quadrant, the value of the angle is $\frac{π}{4}$ .

$θ = \frac{π}{4}$

Step 7

Substitute the values of $θ = \frac{π}{4}$ and $| z | = \sqrt{2}$ .

$\sqrt{2} (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$