Trigonometry Examples

${(3 (\cos (100) + i \sin (100)))}^{3}$

Step 1

Apply the distributive property.

${(3 \cos (100) + 3 (i \sin (100)))}^{3}$

Step 2

Use the Binomial Theorem.

${(3 \cos (100))}^{3} + 3 {(3 \cos (100))}^{2} (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $3 \cos (100)$ .

$3^{3} \cos^{3} (100) + 3 {(3 \cos (100))}^{2} (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.2

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

$27 \cos^{3} (100) + 3 {(3 \cos (100))}^{2} (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.3

Apply the product rule to $3 \cos (100)$ .

$27 \cos^{3} (100) + 3 (3^{2} \cos^{2} (100)) (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.4

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 3.1.4.1

Move $3^{2}$ .

$27 \cos^{3} (100) + 3^{2} \cdot 3 \cos^{2} (100) (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.4.2

Multiply $3^{2}$ by $3$ .

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Step 3.1.4.2.1

Raise $3$ to the power of $1$ .

$27 \cos^{3} (100) + 3^{2} \cdot 3^{1} \cos^{2} (100) (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.4.2.2

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

$27 \cos^{3} (100) + 3^{2 + 1} \cos^{2} (100) (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.4.3

Add $2$ and $1$ .

$27 \cos^{3} (100) + 3^{3} \cos^{2} (100) (3 i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.5

Multiply $3^{3}$ by $3$ by adding the exponents.

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Step 3.1.5.1

Move $3$ .

$27 \cos^{3} (100) + 3 \cdot 3^{3} \cos^{2} (100) (i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.5.2

Multiply $3$ by $3^{3}$ .

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Step 3.1.5.2.1

Raise $3$ to the power of $1$ .

$27 \cos^{3} (100) + 3^{1} \cdot 3^{3} \cos^{2} (100) (i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.5.2.2

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

$27 \cos^{3} (100) + 3^{1 + 3} \cos^{2} (100) (i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.5.3

Add $1$ and $3$ .

$27 \cos^{3} (100) + 3^{4} \cos^{2} (100) (i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.6

Raise $3$ to the power of $4$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) (i \sin (100)) + 3 (3 \cos (100)) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.7

Multiply $3$ by $3$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) + 9 \cos (100) {(3 i \sin (100))}^{2} + {(3 i \sin (100))}^{3}$

Step 3.1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.8.1

Apply the product rule to $3 i \sin (100)$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) + 9 \cos (100) ({(3 i)}^{2} \sin^{2} (100)) + {(3 i \sin (100))}^{3}$

Step 3.1.8.2

Apply the product rule to $3 i$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) + 9 \cos (100) (3^{2} i^{2} \sin^{2} (100)) + {(3 i \sin (100))}^{3}$

Step 3.1.9

Raise $3$ to the power of $2$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) + 9 \cos (100) (9 i^{2} \sin^{2} (100)) + {(3 i \sin (100))}^{3}$

Step 3.1.10

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

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) + 9 \cos (100) (9 \cdot - 1 \sin^{2} (100)) + {(3 i \sin (100))}^{3}$

Step 3.1.11

Multiply $9$ by $- 1$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) + 9 \cos (100) (- 9 \sin^{2} (100)) + {(3 i \sin (100))}^{3}$

Step 3.1.12

Multiply $- 9$ by $9$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + {(3 i \sin (100))}^{3}$

Step 3.1.13

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.13.1

Apply the product rule to $3 i \sin (100)$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + {(3 i)}^{3} \sin^{3} (100)$

Step 3.1.13.2

Apply the product rule to $3 i$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + 3^{3} i^{3} \sin^{3} (100)$

Step 3.1.14

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

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + 27 i^{3} \sin^{3} (100)$

Step 3.1.15

Factor out $i^{2}$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + 27 (i^{2} \cdot i) \sin^{3} (100)$

Step 3.1.16

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

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + 27 (- 1 \cdot i) \sin^{3} (100)$

Step 3.1.17

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

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) + 27 (- i) \sin^{3} (100)$

Step 3.1.18

Multiply $- 1$ by $27$ .

$27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) - 27 i \sin^{3} (100)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Reorder factors in $27 \cos^{3} (100) + 81 \cos^{2} (100) i \sin (100) - 81 \cos (100) \sin^{2} (100) - 27 i \sin^{3} (100)$ .

$27 \cos^{3} (100) + 81 i \cos^{2} (100) \sin (100) - 81 \cos (100) \sin^{2} (100) - 27 i \sin^{3} (100)$

Step 3.2.2

Reorder $27 \cos^{3} (100) + 81 i \cos^{2} (100) \sin (100)$ and $- 81 \cos (100) \sin^{2} (100)$ .

$- 81 \cos (100) \sin^{2} (100) + 27 \cos^{3} (100) + 81 i \cos^{2} (100) \sin (100) - 27 i \sin^{3} (100)$