Trigonometry Examples

$6 (\cos (32) + i \sin (32)) \cdot (5 (\cos (4) + i \sin (4)))$

Step 1

Simplify by multiplying through.

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Step 1.1

Apply the distributive property.

$(6 \cos (32) + 6 (i \sin (32))) \cdot (5 (\cos (4) + i \sin (4)))$

Step 1.2

Apply the distributive property.

$(6 \cos (32) + 6 i \sin (32)) \cdot (5 \cos (4) + 5 (i \sin (4)))$

Step 2

Expand $(6 \cos (32) + 6 i \sin (32)) (5 \cos (4) + 5 i \sin (4))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$6 \cos (32) (5 \cos (4) + 5 i \sin (4)) + 6 i \sin (32) (5 \cos (4) + 5 i \sin (4))$

Step 2.2

Apply the distributive property.

$6 \cos (32) (5 \cos (4)) + 6 \cos (32) (5 i \sin (4)) + 6 i \sin (32) (5 \cos (4) + 5 i \sin (4))$

Step 2.3

Apply the distributive property.

$6 \cos (32) (5 \cos (4)) + 6 \cos (32) (5 i \sin (4)) + 6 i \sin (32) (5 \cos (4)) + 6 i \sin (32) (5 i \sin (4))$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $5$ by $6$ .

$30 \cos (32) \cos (4) + 6 \cos (32) (5 i \sin (4)) + 6 i \sin (32) (5 \cos (4)) + 6 i \sin (32) (5 i \sin (4))$

Step 3.1.2

Multiply $5$ by $6$ .

$30 \cos (32) \cos (4) + 30 \cos (32) (i \sin (4)) + 6 i \sin (32) (5 \cos (4)) + 6 i \sin (32) (5 i \sin (4))$

Step 3.1.3

Multiply $5$ by $6$ .

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 6 i \sin (32) (5 i \sin (4))$

Step 3.1.4

Multiply $6 i \sin (32) (5 i \sin (4))$ .

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Step 3.1.4.1

Multiply $5$ by $6$ .

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 30 i \sin (32) (i \sin (4))$

Step 3.1.4.2

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

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 30 (i^{1} i) \sin (32) \sin (4)$

Step 3.1.4.3

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

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 30 (i^{1} i^{1}) \sin (32) \sin (4)$

Step 3.1.4.4

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

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 30 i^{1 + 1} \sin (32) \sin (4)$

Step 3.1.4.5

Add $1$ and $1$ .

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 30 i^{2} \sin (32) \sin (4)$

Step 3.1.5

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

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) + 30 \cdot - 1 \sin (32) \sin (4)$

Step 3.1.6

Multiply $30$ by $- 1$ .

$30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) - 30 \sin (32) \sin (4)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Reorder factors in $30 \cos (32) \cos (4) + 30 \cos (32) i \sin (4) + 30 i \sin (32) \cos (4) - 30 \sin (32) \sin (4)$ .

$30 \cos (32) \cos (4) + 30 i \cos (32) \sin (4) + 30 i \sin (32) \cos (4) - 30 \sin (32) \sin (4)$

Step 3.2.2

Reorder $30 \cos (32) \cos (4) + 30 i \cos (32) \sin (4) + 30 i \sin (32) \cos (4)$ and $- 30 \sin (32) \sin (4)$ .

$- 30 \sin (32) \sin (4) + 30 \cos (32) \cos (4) + 30 i \cos (32) \sin (4) + 30 i \sin (32) \cos (4)$