Trigonometry Examples

$z = | 7 i |$

Step 1

Simplify $| 7 i |$ .

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Step 1.1

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$z = \sqrt{0^{2} + 7^{2}}$

Step 1.2

Raising $0$ to any positive power yields $0$ .

$z = \sqrt{0 + 7^{2}}$

Step 1.3

Raise $7$ to the power of $2$ .

$z = \sqrt{0 + 49}$

Step 1.4

Add $0$ and $49$ .

$z = \sqrt{49}$

Step 1.5

Rewrite $49$ as $7^{2}$ .

$z = \sqrt{7^{2}}$

Step 1.6

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

$z = 7$

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 $a = 7$ and $b = 0$ .

$| z | = \sqrt{0^{2} + 7^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + 7^{2}}$

Step 5.2

Raise $7$ to the power of $2$ .

$| z | = \sqrt{0 + 49}$

Step 5.3

Add $0$ and $49$ .

$| z | = \sqrt{49}$

Step 5.4

Rewrite $49$ as $7^{2}$ .

$| z | = \sqrt{7^{2}}$

Step 5.5

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

$| z | = 7$

Step 6

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

$θ = \arctan (\frac{0}{7})$

Step 7

Since inverse tangent of $\frac{0}{7}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 8

Substitute the values of $θ = 0$ and $| z | = 7$ .

$7 (\cos (0) + i \sin (0))$

Step 9

Replace the right side of the equation with the trigonometric form.

$z = 7 (\cos (0) + i \sin (0))$