Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Evaluate $\cos (\frac{3 π}{5})$ .

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

Step 1.2

Evaluate $\sin (\frac{3 π}{5})$ .

${(- 0.30901699 + i \cdot 0.95105651)}^{3}$

Step 1.3

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

${(- 0.30901699 + 0.95105651 i)}^{3}$

Step 2

Use the Binomial Theorem.

${(- 0.30901699)}^{3} + 3 {(- 0.30901699)}^{2} (0.95105651 i) + 3 \cdot - 0.30901699 {(0.95105651 i)}^{2} + {(0.95105651 i)}^{3}$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise $- 0.30901699$ to the power of $3$ .

$- 0.02950849 + 3 {(- 0.30901699)}^{2} (0.95105651 i) + 3 \cdot - 0.30901699 {(0.95105651 i)}^{2} + {(0.95105651 i)}^{3}$

Step 3.1.2

Raise $- 0.30901699$ to the power of $2$ .

$- 0.02950849 + 3 \cdot 0.0954915 (0.95105651 i) + 3 \cdot - 0.30901699 {(0.95105651 i)}^{2} + {(0.95105651 i)}^{3}$

Step 3.1.3

Multiply $3$ by $0.0954915$ .

$- 0.02950849 + 0.2864745 (0.95105651 i) + 3 \cdot - 0.30901699 {(0.95105651 i)}^{2} + {(0.95105651 i)}^{3}$

Step 3.1.4

Multiply $0.95105651$ by $0.2864745$ .

$- 0.02950849 + 0.27245344 i + 3 \cdot - 0.30901699 {(0.95105651 i)}^{2} + {(0.95105651 i)}^{3}$

Step 3.1.5

Multiply $3$ by $- 0.30901699$ .

$- 0.02950849 + 0.27245344 i - 0.92705098 {(0.95105651 i)}^{2} + {(0.95105651 i)}^{3}$

Step 3.1.6

Apply the product rule to $0.95105651 i$ .

$- 0.02950849 + 0.27245344 i - 0.92705098 ({0.95105651}^{2} i^{2}) + {(0.95105651 i)}^{3}$

Step 3.1.7

Raise $0.95105651$ to the power of $2$ .

$- 0.02950849 + 0.27245344 i - 0.92705098 (0.90450849 i^{2}) + {(0.95105651 i)}^{3}$

Step 3.1.8

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

$- 0.02950849 + 0.27245344 i - 0.92705098 (0.90450849 \cdot - 1) + {(0.95105651 i)}^{3}$

Step 3.1.9

Multiply $- 0.92705098 (0.90450849 \cdot - 1)$ .

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Step 3.1.9.1

Multiply $0.90450849$ by $- 1$ .

$- 0.02950849 + 0.27245344 i - 0.92705098 \cdot - 0.90450849 + {(0.95105651 i)}^{3}$

Step 3.1.9.2

Multiply $- 0.92705098$ by $- 0.90450849$ .

$- 0.02950849 + 0.27245344 i + 0.83852549 + {(0.95105651 i)}^{3}$

Step 3.1.10

Apply the product rule to $0.95105651 i$ .

$- 0.02950849 + 0.27245344 i + 0.83852549 + {0.95105651}^{3} i^{3}$

Step 3.1.11

Raise $0.95105651$ to the power of $3$ .

$- 0.02950849 + 0.27245344 i + 0.83852549 + 0.8602387 i^{3}$

Step 3.1.12

Factor out $i^{2}$ .

$- 0.02950849 + 0.27245344 i + 0.83852549 + 0.8602387 (i^{2} \cdot i)$

Step 3.1.13

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

$- 0.02950849 + 0.27245344 i + 0.83852549 + 0.8602387 (- 1 \cdot i)$

Step 3.1.14

Rewrite $- 1 i$ as $- i$ .

$- 0.02950849 + 0.27245344 i + 0.83852549 + 0.8602387 (- i)$

Step 3.1.15

Multiply $- 1$ by $0.8602387$ .

$- 0.02950849 + 0.27245344 i + 0.83852549 - 0.8602387 i$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $- 0.02950849$ and $0.83852549$ .

$0.80901699 + 0.27245344 i - 0.8602387 i$

Step 3.2.2

Subtract $0.8602387 i$ from $0.27245344 i$ .

$0.80901699 - 0.58778525 i$

Step 4

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 5

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 6

Substitute the actual values of $a = 0.80901699$ and $b = - 0.58778525$ .

$| z | = \sqrt{{(- 0.58778525)}^{2} + {0.80901699}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

Raise $- 0.58778525$ to the power of $2$ .

$| z | = \sqrt{0.3454915 + {0.80901699}^{2}}$

Step 7.2

Raise $0.80901699$ to the power of $2$ .

$| z | = \sqrt{0.3454915 + 0.65450849}$

Step 7.3

Add $0.3454915$ and $0.65450849$ .

$| z | = \sqrt{1}$

Step 7.4

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

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

Step 7.5

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

$| z | = 1$

Step 8

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

$θ = \arctan (\frac{- 0.58778525}{0.80901699})$

Step 9

Since inverse tangent of $\frac{- 0.58778525}{0.80901699}$ produces an angle in the fourth quadrant, the value of the angle is $- \frac{π}{5}$ .

$θ = - \frac{π}{5}$

Step 10

Substitute the values of $θ = - \frac{π}{5}$ and $| z | = 1$ .

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