Trigonometry Examples

$8 i (8 - 3 i)$

Step 1

Apply the distributive property.

$8 i \cdot 8 + 8 i (- 3 i)$

Step 2

Multiply $8$ by $8$ .

$64 i + 8 i (- 3 i)$

Step 3

Multiply $8 i (- 3 i)$ .

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Step 3.1

Multiply $- 3$ by $8$ .

$64 i - 24 i i$

Step 3.2

Raise $i$ to the power of $1$ .

$64 i - 24 (i^{1} i)$

Step 3.3

Raise $i$ to the power of $1$ .

$64 i - 24 (i^{1} i^{1})$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$64 i - 24 i^{1 + 1}$

Step 3.5

Add $1$ and $1$ .

$64 i - 24 i^{2}$

Step 4

Simplify each term.

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Step 4.1

Rewrite $i^{2}$ as $- 1$ .

$64 i - 24 \cdot - 1$

Step 4.2

Multiply $- 24$ by $- 1$ .

$64 i + 24$

Step 5

Reorder $64 i$ and $24$ .

$24 + 64 i$

Step 6

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 7

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 8

Substitute the actual values of $a = 24$ and $b = 64$ .

$| z | = \sqrt{64^{2} + 24^{2}}$

Step 9

Find $| z |$ .

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Step 9.1

Raise $64$ to the power of $2$ .

$| z | = \sqrt{4096 + 24^{2}}$

Step 9.2

Raise $24$ to the power of $2$ .

$| z | = \sqrt{4096 + 576}$

Step 9.3

Add $4096$ and $576$ .

$| z | = \sqrt{4672}$

Step 9.4

Rewrite $4672$ as $8^{2} \cdot 73$ .

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Step 9.4.1

Factor $64$ out of $4672$ .

$| z | = \sqrt{64 (73)}$

Step 9.4.2

Rewrite $64$ as $8^{2}$ .

$| z | = \sqrt{8^{2} \cdot 73}$

Step 9.5

Pull terms out from under the radical.

$| z | = 8 \sqrt{73}$

Step 10

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

$θ = \arctan (\frac{64}{24})$

Step 11

Since inverse tangent of $\frac{64}{24}$ produces an angle in the first quadrant, the value of the angle is $1.21202565$ .

$θ = 1.21202565$

Step 12

Substitute the values of $θ = 1.21202565$ and $| z | = 8 \sqrt{73}$ .

$8 \sqrt{73} (\cos (1.21202565) + i \sin (1.21202565))$