Algebra Examples

$\sin (- 6 x)$

Step 1

Simplify terms.

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Step 1.1

Factor $2$ out of $- 6 x$ .

$\sin (2 (- 3 x))$

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Factor $3$ out of $- 3 x$ .

$2 \sin (3 (- x)) \cos (- 3 x)$

Step 1.2.2

Apply the sine triple-angle identity.

$2 (- 4 \sin^{3} (- x) + 3 \sin (- x)) \cos (- 3 x)$

Step 1.3

Simplify each term.

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Step 1.3.1

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$2 (- 4 {(- \sin (x))}^{3} + 3 \sin (- x)) \cos (- 3 x)$

Step 1.3.2

Apply the product rule to $- \sin (x)$ .

$2 (- 4 ({(- 1)}^{3} \sin^{3} (x)) + 3 \sin (- x)) \cos (- 3 x)$

Step 1.3.3

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

$2 (- 4 (- \sin^{3} (x)) + 3 \sin (- x)) \cos (- 3 x)$

Step 1.3.4

Multiply $- 1$ by $- 4$ .

$2 (4 \sin^{3} (x) + 3 \sin (- x)) \cos (- 3 x)$

Step 1.3.5

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$2 (4 \sin^{3} (x) + 3 (- \sin (x))) \cos (- 3 x)$

Step 1.3.6

Multiply $- 1$ by $3$ .

$2 (4 \sin^{3} (x) - 3 \sin (x)) \cos (- 3 x)$

Step 1.4

Simplify terms.

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Step 1.4.1

Apply the distributive property.

$(2 (4 \sin^{3} (x)) + 2 (- 3 \sin (x))) \cos (- 3 x)$

Step 1.4.2

Multiply.

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Step 1.4.2.1

Multiply $4$ by $2$ .

$(8 \sin^{3} (x) + 2 (- 3 \sin (x))) \cos (- 3 x)$

Step 1.4.2.2

Multiply $- 3$ by $2$ .

$(8 \sin^{3} (x) - 6 \sin (x)) \cos (- 3 x)$

Step 1.4.3

Factor $3$ out of $- 3 x$ .

$(8 \sin^{3} (x) - 6 \sin (x)) \cos (3 (- x))$

Step 1.4.4

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

$(8 \sin^{3} (x) - 6 \sin (x)) (4 \cos^{3} (- x) - 3 \cos (- x))$

Step 1.5

Simplify each term.

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Step 1.5.1

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$(8 \sin^{3} (x) - 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (- x))$

Step 1.5.2

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$(8 \sin^{3} (x) - 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x))$

Step 2

Expand $(8 \sin^{3} (x) - 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$8 \sin^{3} (x) (4 \cos^{3} (x) - 3 \cos (x)) - 6 \sin (x) (4 \cos^{3} (x) - 3 \cos (x))$

Step 2.2

Apply the distributive property.

$8 \sin^{3} (x) (4 \cos^{3} (x)) + 8 \sin^{3} (x) (- 3 \cos (x)) - 6 \sin (x) (4 \cos^{3} (x) - 3 \cos (x))$

Step 2.3

Apply the distributive property.

$8 \sin^{3} (x) (4 \cos^{3} (x)) + 8 \sin^{3} (x) (- 3 \cos (x)) - 6 \sin (x) (4 \cos^{3} (x)) - 6 \sin (x) (- 3 \cos (x))$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$8 \cdot 4 \sin^{3} (x) \cos^{3} (x) + 8 \sin^{3} (x) (- 3 \cos (x)) - 6 \sin (x) (4 \cos^{3} (x)) - 6 \sin (x) (- 3 \cos (x))$

Step 3.1.2

Multiply $8$ by $4$ .

$32 \sin^{3} (x) \cos^{3} (x) + 8 \sin^{3} (x) (- 3 \cos (x)) - 6 \sin (x) (4 \cos^{3} (x)) - 6 \sin (x) (- 3 \cos (x))$

Step 3.1.3

Multiply $- 3$ by $8$ .

$32 \sin^{3} (x) \cos^{3} (x) - 24 \sin^{3} (x) \cos (x) - 6 \sin (x) (4 \cos^{3} (x)) - 6 \sin (x) (- 3 \cos (x))$

Step 3.1.4

Multiply $4$ by $- 6$ .

$32 \sin^{3} (x) \cos^{3} (x) - 24 \sin^{3} (x) \cos (x) - 24 \sin (x) \cos^{3} (x) - 6 \sin (x) (- 3 \cos (x))$

Step 3.1.5

Multiply $- 3$ by $- 6$ .

$32 \sin^{3} (x) \cos^{3} (x) - 24 \sin^{3} (x) \cos (x) - 24 \sin (x) \cos^{3} (x) + 18 \sin (x) \cos (x)$

Step 3.2

Simplify with factoring out.

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Step 3.2.1

Factor $2 \sin (x) \cos (x)$ out of $32 \sin^{3} (x) \cos^{3} (x) - 24 \sin^{3} (x) \cos (x) - 24 \sin (x) \cos^{3} (x) + 18 \sin (x) \cos (x)$ .

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Step 3.2.1.1

Factor $2 \sin (x) \cos (x)$ out of $32 \sin^{3} (x) \cos^{3} (x)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x)) - 24 \sin^{3} (x) \cos (x) - 24 \sin (x) \cos^{3} (x) + 18 \sin (x) \cos (x)$

Step 3.2.1.2

Factor $2 \sin (x) \cos (x)$ out of $- 24 \sin^{3} (x) \cos (x)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \sin^{2} (x)) - 24 \sin (x) \cos^{3} (x) + 18 \sin (x) \cos (x)$

Step 3.2.1.3

Factor $2 \sin (x) \cos (x)$ out of $- 24 \sin (x) \cos^{3} (x)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \cos^{2} (x)) + 18 \sin (x) \cos (x)$

Step 3.2.1.4

Factor $2 \sin (x) \cos (x)$ out of $18 \sin (x) \cos (x)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (9)$

Step 3.2.1.5

Factor $2 \sin (x) \cos (x)$ out of $2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \sin^{2} (x))$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (9)$

Step 3.2.1.6

Factor $2 \sin (x) \cos (x)$ out of $2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (- 12 \cos^{2} (x))$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 \sin^{2} (x) - 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (9)$

Step 3.2.1.7

Factor $2 \sin (x) \cos (x)$ out of $2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 \sin^{2} (x) - 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (9)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 \sin^{2} (x) - 12 \cos^{2} (x) + 9)$

Step 3.2.2

Factor $- 12$ out of $- 12 \sin^{2} (x)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 (\sin^{2} (x)) - 12 \cos^{2} (x) + 9)$

Step 3.2.3

Factor $- 12$ out of $- 12 \cos^{2} (x)$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 (\sin^{2} (x)) - 12 (\cos^{2} (x)) + 9)$

Step 3.2.4

Factor $- 12$ out of $- 12 (\sin^{2} (x)) - 12 (\cos^{2} (x))$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 (\sin^{2} (x) + \cos^{2} (x)) + 9)$

Step 4

Apply pythagorean identity.

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 \cdot 1 + 9)$

Step 5

Simplify by multiplying through.

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Step 5.1

Multiply $- 12$ by $1$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 12 + 9)$

Step 5.2

Add $- 12$ and $9$ .

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x) - 3)$

Step 5.3

Apply the distributive property.

$2 \sin (x) \cos (x) (16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot - 3$

Step 6

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 6.1

Move $\sin^{2} (x)$ .

$2 (\sin^{2} (x) \sin (x)) \cos (x) (16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot - 3$

Step 6.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 6.2.1

Raise $\sin (x)$ to the power of $1$ .

$2 (\sin^{2} (x) \sin^{1} (x)) \cos (x) (16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot - 3$

Step 6.2.2

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

$2 {\sin (x)}^{2 + 1} \cos (x) (16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot - 3$

Step 6.3

Add $2$ and $1$ .

$2 \sin^{3} (x) \cos (x) (16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot - 3$

Step 7

Multiply $- 3$ by $2$ .

$2 \sin^{3} (x) \cos (x) (16 \cos^{2} (x)) - 6 \sin (x) \cos (x)$

Step 8

Simplify each term.

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Step 8.1

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 8.1.1

Move $\cos^{2} (x)$ .

$2 \sin^{3} (x) (\cos^{2} (x) \cos (x)) \cdot 16 - 6 \sin (x) \cos (x)$

Step 8.1.2

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 8.1.2.1

Raise $\cos (x)$ to the power of $1$ .

$2 \sin^{3} (x) (\cos^{2} (x) \cos^{1} (x)) \cdot 16 - 6 \sin (x) \cos (x)$

Step 8.1.2.2

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

$2 \sin^{3} (x) {\cos (x)}^{2 + 1} \cdot 16 - 6 \sin (x) \cos (x)$

Step 8.1.3

Add $2$ and $1$ .

$2 \sin^{3} (x) \cos^{3} (x) \cdot 16 - 6 \sin (x) \cos (x)$

Step 8.2

Multiply $16$ by $2$ .

$32 \sin^{3} (x) \cos^{3} (x) - 6 \sin (x) \cos (x)$