Trigonometry Examples

${(6 (\cos (110) + i \sin (110)))}^{3}$

Step 1

Apply the distributive property.

${(6 \cos (110) + 6 (i \sin (110)))}^{3}$

Step 2

Use the Binomial Theorem.

${(6 \cos (110))}^{3} + 3 {(6 \cos (110))}^{2} (6 i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

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 $6 \cos (110)$ .

$6^{3} \cos^{3} (110) + 3 {(6 \cos (110))}^{2} (6 i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.2

Raise $6$ to the power of $3$ .

$216 \cos^{3} (110) + 3 {(6 \cos (110))}^{2} (6 i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.3

Apply the product rule to $6 \cos (110)$ .

$216 \cos^{3} (110) + 3 (6^{2} \cos^{2} (110)) (6 i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.4

Multiply $6^{2}$ by $6$ by adding the exponents.

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Step 3.1.4.1

Move $6$ .

$216 \cos^{3} (110) + 3 (6 \cdot 6^{2} \cos^{2} (110)) (i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.4.2

Multiply $6$ by $6^{2}$ .

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Step 3.1.4.2.1

Raise $6$ to the power of $1$ .

$216 \cos^{3} (110) + 3 (6^{1} \cdot 6^{2} \cos^{2} (110)) (i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.4.2.2

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

$216 \cos^{3} (110) + 3 (6^{1 + 2} \cos^{2} (110)) (i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.4.3

Add $1$ and $2$ .

$216 \cos^{3} (110) + 3 (6^{3} \cos^{2} (110)) (i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.5

Raise $6$ to the power of $3$ .

$216 \cos^{3} (110) + 3 (216 \cos^{2} (110)) (i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.6

Multiply $216$ by $3$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) (i \sin (110)) + 3 (6 \cos (110)) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

Step 3.1.7

Multiply $6$ by $3$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) + 18 \cos (110) {(6 i \sin (110))}^{2} + {(6 i \sin (110))}^{3}$

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 $6 i \sin (110)$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) + 18 \cos (110) ({(6 i)}^{2} \sin^{2} (110)) + {(6 i \sin (110))}^{3}$

Step 3.1.8.2

Apply the product rule to $6 i$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) + 18 \cos (110) (6^{2} i^{2} \sin^{2} (110)) + {(6 i \sin (110))}^{3}$

Step 3.1.9

Raise $6$ to the power of $2$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) + 18 \cos (110) (36 i^{2} \sin^{2} (110)) + {(6 i \sin (110))}^{3}$

Step 3.1.10

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

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) + 18 \cos (110) (36 \cdot - 1 \sin^{2} (110)) + {(6 i \sin (110))}^{3}$

Step 3.1.11

Multiply $36$ by $- 1$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) + 18 \cos (110) (- 36 \sin^{2} (110)) + {(6 i \sin (110))}^{3}$

Step 3.1.12

Multiply $- 36$ by $18$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + {(6 i \sin (110))}^{3}$

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 $6 i \sin (110)$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + {(6 i)}^{3} \sin^{3} (110)$

Step 3.1.13.2

Apply the product rule to $6 i$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + 6^{3} i^{3} \sin^{3} (110)$

Step 3.1.14

Raise $6$ to the power of $3$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + 216 i^{3} \sin^{3} (110)$

Step 3.1.15

Factor out $i^{2}$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + 216 (i^{2} \cdot i) \sin^{3} (110)$

Step 3.1.16

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

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + 216 (- 1 \cdot i) \sin^{3} (110)$

Step 3.1.17

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

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) + 216 (- i) \sin^{3} (110)$

Step 3.1.18

Multiply $- 1$ by $216$ .

$216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) - 216 i \sin^{3} (110)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Reorder factors in $216 \cos^{3} (110) + 648 \cos^{2} (110) i \sin (110) - 648 \cos (110) \sin^{2} (110) - 216 i \sin^{3} (110)$ .

$216 \cos^{3} (110) + 648 i \cos^{2} (110) \sin (110) - 648 \cos (110) \sin^{2} (110) - 216 i \sin^{3} (110)$

Step 3.2.2

Reorder $216 \cos^{3} (110) + 648 i \cos^{2} (110) \sin (110)$ and $- 648 \cos (110) \sin^{2} (110)$ .

$- 648 \cos (110) \sin^{2} (110) + 216 \cos^{3} (110) + 648 i \cos^{2} (110) \sin (110) - 216 i \sin^{3} (110)$