Precalculus Examples

$(\frac{3}{5} (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{3}{5} (\frac{1}{2} + i \sin (\frac{π}{3})) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{3}{5} (\frac{1}{2} + i \frac{\sqrt{3}}{2}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 1.3

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$\frac{3}{5} (\frac{1}{2} + \frac{i \sqrt{3}}{2}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 2

Apply the distributive property.

$(\frac{3}{5} \cdot \frac{1}{2} + \frac{3}{5} \cdot \frac{i \sqrt{3}}{2}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 3

Multiply $\frac{3}{5} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 3.1

Multiply $\frac{3}{5}$ by $\frac{1}{2}$ .

$(\frac{3}{5 \cdot 2} + \frac{3}{5} \cdot \frac{i \sqrt{3}}{2}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 3.2

Multiply $5$ by $2$ .

$(\frac{3}{10} + \frac{3}{5} \cdot \frac{i \sqrt{3}}{2}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 4

Multiply $\frac{3}{5} \cdot \frac{i \sqrt{3}}{2}$ .

Tap for more steps...

Step 4.1

Multiply $\frac{3}{5}$ by $\frac{i \sqrt{3}}{2}$ .

$(\frac{3}{10} + \frac{3 (i \sqrt{3})}{5 \cdot 2}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 4.2

Multiply $5$ by $2$ .

$(\frac{3}{10} + \frac{3 (i \sqrt{3})}{10}) (5 (\cos (\frac{3 π}{5}) + i \sin (\frac{3 π}{5})))$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

Evaluate $\cos (\frac{3 π}{5})$ .

$(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (5 (- 0.30901699 + i \sin (\frac{3 π}{5})))$

Step 5.1.2

Evaluate $\sin (\frac{3 π}{5})$ .

$(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (5 (- 0.30901699 + i \cdot 0.95105651))$

Step 5.1.3

Move $0.95105651$ to the left of $i$ .

$(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (5 (- 0.30901699 + 0.95105651 i))$

Step 5.2

Simplify by multiplying through.

Tap for more steps...

Step 5.2.1

Apply the distributive property.

$(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (5 \cdot - 0.30901699 + 5 (0.95105651 i))$

Step 5.2.2

Multiply.

Tap for more steps...

Step 5.2.2.1

Multiply $5$ by $- 0.30901699$ .

$(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (- 1.54508497 + 5 (0.95105651 i))$

Step 5.2.2.2

Multiply $0.95105651$ by $5$ .

$(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (- 1.54508497 + 4.75528258 i)$

Step 6

Expand $(\frac{3}{10} + \frac{3 i \sqrt{3}}{10}) (- 1.54508497 + 4.75528258 i)$ using the FOIL Method.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{3}{10} (- 1.54508497 + 4.75528258 i) + \frac{3 i \sqrt{3}}{10} (- 1.54508497 + 4.75528258 i)$

Step 6.2

Apply the distributive property.

$\frac{3}{10} \cdot - 1.54508497 + \frac{3}{10} (4.75528258 i) + \frac{3 i \sqrt{3}}{10} (- 1.54508497 + 4.75528258 i)$

Step 6.3

Apply the distributive property.

$\frac{3}{10} \cdot - 1.54508497 + \frac{3}{10} (4.75528258 i) + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7

Simplify and combine like terms.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply $\frac{3}{10} \cdot - 1.54508497$ .

Tap for more steps...

Step 7.1.1.1

Combine $\frac{3}{10}$ and $- 1.54508497$ .

$\frac{3 \cdot - 1.54508497}{10} + \frac{3}{10} (4.75528258 i) + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.1.2

Multiply $3$ by $- 1.54508497$ .

$\frac{- 4.63525491}{10} + \frac{3}{10} (4.75528258 i) + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.2

Divide $- 4.63525491$ by $10$ .

$- 0.46352549 + \frac{3}{10} (4.75528258 i) + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.3

Multiply $\frac{3}{10} (4.75528258 i)$ .

Tap for more steps...

Step 7.1.3.1

Combine $4.75528258$ and $\frac{3}{10}$ .

$- 0.46352549 + \frac{4.75528258 \cdot 3}{10} i + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.3.2

Multiply $4.75528258$ by $3$ .

$- 0.46352549 + \frac{14.26584774}{10} i + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.3.3

Combine $\frac{14.26584774}{10}$ and $i$ .

$- 0.46352549 + \frac{14.26584774 i}{10} + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.4

Factor $14.26584774$ out of $14.26584774 i$ .

$- 0.46352549 + \frac{14.26584774 (i)}{10} + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.5

Factor $10$ out of $10$ .

$- 0.46352549 + \frac{14.26584774 (i)}{10 (1)} + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.6

Separate fractions.

$- 0.46352549 + \frac{14.26584774}{10} \cdot \frac{i}{1} + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.7

Divide $14.26584774$ by $10$ .

$- 0.46352549 + 1.42658477 \frac{i}{1} + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.8

Divide $i$ by $1$ .

$- 0.46352549 + 1.42658477 i + \frac{3 i \sqrt{3}}{10} \cdot - 1.54508497 + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.9

Multiply $\frac{3 i \sqrt{3}}{10} \cdot - 1.54508497$ .

Tap for more steps...

Step 7.1.9.1

Combine $\frac{3 i \sqrt{3}}{10}$ and $- 1.54508497$ .

$- 0.46352549 + 1.42658477 i + \frac{3 i \sqrt{3} \cdot - 1.54508497}{10} + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.9.2

Multiply $- 1.54508497$ by $3$ .

$- 0.46352549 + 1.42658477 i + \frac{- 4.63525491 i \sqrt{3}}{10} + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.9.3

Multiply $\sqrt{3}$ by $- 4.63525491$ .

$- 0.46352549 + 1.42658477 i + \frac{- 8.02849701 i}{10} + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.10

Move the negative in front of the fraction.

$- 0.46352549 + 1.42658477 i - \frac{(8.02849701) i}{10} + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.11

Factor $8.02849701$ out of $8.02849701 i$ .

$- 0.46352549 + 1.42658477 i - \frac{8.02849701 (i)}{10} + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.12

Factor $10$ out of $10$ .

$- 0.46352549 + 1.42658477 i - \frac{8.02849701 (i)}{10 (1)} + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.13

Separate fractions.

$- 0.46352549 + 1.42658477 i - (\frac{8.02849701}{10} \cdot \frac{i}{1}) + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.14

Divide $8.02849701$ by $10$ .

$- 0.46352549 + 1.42658477 i - (0.8028497 \frac{i}{1}) + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.15

Divide $i$ by $1$ .

$- 0.46352549 + 1.42658477 i - (0.8028497 i) + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.16

Multiply $0.8028497$ by $- 1$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{3 i \sqrt{3}}{10} (4.75528258 i)$

Step 7.1.17

Multiply $\frac{3 i \sqrt{3}}{10} (4.75528258 i)$ .

Tap for more steps...

Step 7.1.17.1

Combine $4.75528258$ and $\frac{3 i \sqrt{3}}{10}$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{4.75528258 (3 i \sqrt{3})}{10} i$

Step 7.1.17.2

Multiply $3$ by $4.75528258$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{14.26584774 (i \sqrt{3})}{10} i$

Step 7.1.17.3

Multiply $\sqrt{3}$ by $14.26584774$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 i}{10} i$

Step 7.1.17.4

Combine $\frac{24.7091731 i}{10}$ and $i$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 i i}{10}$

Step 7.1.17.5

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

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 (i^{1} i)}{10}$

Step 7.1.17.6

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

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 (i^{1} i^{1})}{10}$

Step 7.1.17.7

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

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 i^{1 + 1}}{10}$

Step 7.1.17.8

Add $1$ and $1$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 i^{2}}{10}$

Step 7.1.18

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

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{24.7091731 \cdot - 1}{10}$

Step 7.1.19

Multiply $24.7091731$ by $- 1$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i + \frac{- 24.7091731}{10}$

Step 7.1.20

Divide $- 24.7091731$ by $10$ .

$- 0.46352549 + 1.42658477 i - 0.8028497 i - 2.47091731$

Step 7.2

Subtract $2.47091731$ from $- 0.46352549$ .

$- 2.9344428 + 1.42658477 i - 0.8028497 i$

Step 7.3

Subtract $0.8028497 i$ from $1.42658477 i$ .

$- 2.9344428 + 0.62373507 i$

Step 8

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 9

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 10

Substitute the actual values of $a = - 2.9344428$ and $b = 0.62373507$ .

$| z | = \sqrt{{0.62373507}^{2} + {(- 2.9344428)}^{2}}$

Step 11

Find $| z |$ .

Tap for more steps...

Step 11.1

Raise $0.62373507$ to the power of $2$ .

$| z | = \sqrt{0.38904544 + {(- 2.9344428)}^{2}}$

Step 11.2

Raise $- 2.9344428$ to the power of $2$ .

$| z | = \sqrt{0.38904544 + 8.61095455}$

Step 11.3

Add $0.38904544$ and $8.61095455$ .

$| z | = \sqrt{9}$

Step 11.4

Rewrite $9$ as $3^{2}$ .

$| z | = \sqrt{3^{2}}$

Step 11.5

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

$| z | = 3$

Step 12

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

$θ = \arctan (\frac{0.62373507}{- 2.9344428})$

Step 13

Since inverse tangent of $\frac{0.62373507}{- 2.9344428}$ produces an angle in the second quadrant, the value of the angle is $2.93215314$ .

$θ = 2.93215314$

Step 14

Substitute the values of $θ = 2.93215314$ and $| z | = 3$ .

$3 (\cos (2.93215314) + i \sin (2.93215314))$