Trigonometry Examples

$\sin (6 x)$

Step 1

Factor $2$ out of $6 x$ .

$\sin (2 (3 x))$

Step 2

Apply the sine triple-angle identity.

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

Step 3

Apply the distributive property.

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

Step 4

Multiply.

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Step 4.1

Multiply $- 4$ by $2$ .

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

Step 4.2

Multiply $3$ by $2$ .

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

Step 5

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 6

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

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Step 6.1

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 6.2

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 6.3

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 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

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 7.1.2

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 7.1.3

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 7.1.4

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 7.1.5

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 7.2

Simplify with factoring out.

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Step 7.2.1

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 7.2.1.1

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 7.2.1.2

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 7.2.1.3

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 7.2.1.4

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 7.2.1.5

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 7.2.1.6

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 7.2.1.7

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 7.2.2

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 7.2.3

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 7.2.4

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 8

Apply pythagorean identity.

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

Step 9

Simplify by multiplying through.

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Step 9.1

Multiply $12$ by $1$ .

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

Step 9.2

Subtract $9$ from $12$ .

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

Step 9.3

Apply the distributive property.

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

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 10.2.2

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 10.3

Add $2$ and $1$ .

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

Step 11

Multiply $3$ by $2$ .

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

Step 12

Simplify each term.

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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

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Step 12.1.2.1

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 12.1.2.2

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 12.1.3

Add $2$ and $1$ .

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

Step 12.2

Multiply $- 16$ by $2$ .

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