Trigonometry Examples

sin (3 x) = 3 sin (x) - 4 {sin}^{3} (x)

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

Step 1

Start on the right side.

3 sin (x) - 4 {sin}^{3} (x)

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

Step 2

Factor

sin (x)

$\sin (x)$ out of

3 sin (x) - 4 {sin}^{3} (x)

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

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Step 2.1

Factor

sin (x)

$\sin (x)$ out of

3 sin (x)

$3 \sin (x)$ .

sin (x) \cdot 3 - 4 {sin}^{3} (x)

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

Step 2.2

Factor

sin (x)

$\sin (x)$ out of

- 4 {sin}^{3} (x)

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

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

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

Step 2.3

Factor

sin (x)

$\sin (x)$ out of

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

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

sin (x) (3 - 4 {sin}^{2} (x))

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

sin (x) (3 - 4 {sin}^{2} (x))

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

Step 3

Apply Pythagorean identity in reverse.

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

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

Step 4

Apply the distributive property.

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

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

Step 5

Simplify each term.

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

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

Step 6

Apply the distributive property.

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

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

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Move

3

$3$ to the left of

sin (x)

$\sin (x)$ .

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

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

Step 7.1.2

Move

- 4

$- 4$ to the left of

sin (x)

$\sin (x)$ .

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

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

Step 7.1.3

Move

4

$4$ to the left of

sin (x)

$\sin (x)$ .

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

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

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

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

Step 7.2

Subtract

4 sin (x)

$4 \sin (x)$ from

3 sin (x)

$3 \sin (x)$ .

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

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

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

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

Step 8

Apply Pythagorean identity in reverse.

- sin (x) + 4 sin (x) (1 - {sin}^{2} (x))

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

Step 9

Factor.

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Step 9.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

- sin (x) + 4 sin (x) (1^{2} - {sin}^{2} (x))

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

Step 9.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 1

$a = 1$ and

b = sin (x)

$b = \sin (x)$ .

- sin (x) + 4 sin (x) ((1 + sin (x)) (1 - sin (x)))

$- \sin (x) + 4 \sin (x) ((1 + \sin (x)) (1 - \sin (x)))$

Step 9.3

Remove parentheses.

- sin (x) + 4 sin (x) (1 + sin (x)) (1 - sin (x))

$- \sin (x) + 4 \sin (x) (1 + \sin (x)) (1 - \sin (x))$

Step 9.4

Factor

sin (x)

$\sin (x)$ out of

- sin (x) + 4 sin (x) (1 + sin (x)) (1 - sin (x))

$- \sin (x) + 4 \sin (x) (1 + \sin (x)) (1 - \sin (x))$ .

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Step 9.4.1

Factor

sin (x)

$\sin (x)$ out of

- sin (x)

$- \sin (x)$ .

sin (x) \cdot - 1 + 4 sin (x) (1 + sin (x)) (1 - sin (x))

$\sin (x) \cdot - 1 + 4 \sin (x) (1 + \sin (x)) (1 - \sin (x))$

Step 9.4.2

Factor

sin (x)

$\sin (x)$ out of

4 sin (x) (1 + sin (x)) (1 - sin (x))

$4 \sin (x) (1 + \sin (x)) (1 - \sin (x))$ .

sin (x) \cdot - 1 + sin (x) ((4 (1 + sin (x))) (1 - sin (x)))

$\sin (x) \cdot - 1 + \sin (x) ((4 (1 + \sin (x))) (1 - \sin (x)))$

Step 9.4.3

Factor

sin (x)

$\sin (x)$ out of

sin (x) \cdot - 1 + sin (x) ((4 (1 + sin (x))) (1 - sin (x)))

$\sin (x) \cdot - 1 + \sin (x) ((4 (1 + \sin (x))) (1 - \sin (x)))$ .

sin (x) (- 1 + (4 (1 + sin (x))) (1 - sin (x)))

$\sin (x) (- 1 + (4 (1 + \sin (x))) (1 - \sin (x)))$

sin (x) (- 1 + (4 (1 + sin (x))) (1 - sin (x)))

$\sin (x) (- 1 + (4 (1 + \sin (x))) (1 - \sin (x)))$

Step 9.5

Apply the distributive property.

sin (x) (- 1 + (4 \cdot 1 + 4 sin (x)) (1 - sin (x)))

$\sin (x) (- 1 + (4 \cdot 1 + 4 \sin (x)) (1 - \sin (x)))$

Step 9.6

Multiply

4

$4$ by

1

$1$ .

sin (x) (- 1 + (4 + 4 sin (x)) (1 - sin (x)))

$\sin (x) (- 1 + (4 + 4 \sin (x)) (1 - \sin (x)))$

Step 9.7

Expand

(4 + 4 sin (x)) (1 - sin (x))

$(4 + 4 \sin (x)) (1 - \sin (x))$ using the FOIL Method.

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Step 9.7.1

Apply the distributive property.

sin (x) (- 1 + 4 (1 - sin (x)) + 4 sin (x) (1 - sin (x)))

$\sin (x) (- 1 + 4 (1 - \sin (x)) + 4 \sin (x) (1 - \sin (x)))$

Step 9.7.2

Apply the distributive property.

sin (x) (- 1 + 4 \cdot 1 + 4 (- sin (x)) + 4 sin (x) (1 - sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 + 4 (- \sin (x)) + 4 \sin (x) (1 - \sin (x)))$

Step 9.7.3

Apply the distributive property.

sin (x) (- 1 + 4 \cdot 1 + 4 (- sin (x)) + 4 sin (x) \cdot 1 + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 + 4 (- \sin (x)) + 4 \sin (x) \cdot 1 + 4 \sin (x) (- \sin (x)))$

sin (x) (- 1 + 4 \cdot 1 + 4 (- sin (x)) + 4 sin (x) \cdot 1 + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 + 4 (- \sin (x)) + 4 \sin (x) \cdot 1 + 4 \sin (x) (- \sin (x)))$

Step 9.8

Combine the opposite terms in

4 \cdot 1 + 4 (- sin (x)) + 4 sin (x) \cdot 1 + 4 sin (x) (- sin (x))

$4 \cdot 1 + 4 (- \sin (x)) + 4 \sin (x) \cdot 1 + 4 \sin (x) (- \sin (x))$ .

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Step 9.8.1

Reorder the factors in the terms

4 (- sin (x))

$4 (- \sin (x))$ and

4 sin (x) \cdot 1

$4 \sin (x) \cdot 1$ .

sin (x) (- 1 + 4 \cdot 1 - 1 \cdot 4 sin (x) + 1 \cdot 4 sin (x) + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 - 1 \cdot 4 \sin (x) + 1 \cdot 4 \sin (x) + 4 \sin (x) (- \sin (x)))$

Step 9.8.2

Add

- 1 \cdot 4 sin (x)

$- 1 \cdot 4 \sin (x)$ and

1 \cdot 4 sin (x)

$1 \cdot 4 \sin (x)$ .

sin (x) (- 1 + 4 \cdot 1 + 0 + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 + 0 + 4 \sin (x) (- \sin (x)))$

Step 9.8.3

Add

4 \cdot 1

$4 \cdot 1$ and

0

$0$ .

sin (x) (- 1 + 4 \cdot 1 + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 + 4 \sin (x) (- \sin (x)))$

sin (x) (- 1 + 4 \cdot 1 + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 \cdot 1 + 4 \sin (x) (- \sin (x)))$

Step 9.9

Simplify each term.

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Step 9.9.1

Multiply

4

$4$ by

1

$1$ .

sin (x) (- 1 + 4 + 4 sin (x) (- sin (x)))

$\sin (x) (- 1 + 4 + 4 \sin (x) (- \sin (x)))$

Step 9.9.2

Multiply

4 sin (x) (- sin (x))

$4 \sin (x) (- \sin (x))$ .

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Step 9.9.2.1

Multiply

- 1

$- 1$ by

4

$4$ .

sin (x) (- 1 + 4 - 4 sin (x) sin (x))

$\sin (x) (- 1 + 4 - 4 \sin (x) \sin (x))$

Step 9.9.2.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

sin (x) (- 1 + 4 - 4 ({sin}^{1} (x) sin (x)))

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

Step 9.9.2.3

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

sin (x) (- 1 + 4 - 4 ({sin}^{1} (x) {sin}^{1} (x)))

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

Step 9.9.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

sin (x) (- 1 + 4 - 4 {sin (x)}^{1 + 1})

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

Step 9.9.2.5

Add

1

$1$ and

1

$1$ .

sin (x) (- 1 + 4 - 4 {sin}^{2} (x))

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

sin (x) (- 1 + 4 - 4 {sin}^{2} (x))

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

sin (x) (- 1 + 4 - 4 {sin}^{2} (x))

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

Step 9.10

Factor

4

$4$ out of

4

$4$ .

sin (x) (- 1 + 4 (1) - 4 {sin}^{2} (x))

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

Step 9.11

Factor

4

$4$ out of

- 4 {sin}^{2} (x)

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

sin (x) (- 1 + 4 (1) + 4 (- {sin}^{2} (x)))

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

Step 9.12

Factor

4

$4$ out of

4 (1) + 4 (- {sin}^{2} (x))

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

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

Step 9.13

Apply pythagorean identity.

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

Step 9.14

Factor.

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Step 9.14.1

Rewrite

$- 1 + 4 \cos^{2} (x)$ in a factored form.

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Step 9.14.1.1

Rewrite

$4 \cos^{2} (x)$ as

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

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

Step 9.14.1.2

Rewrite

$1$ as

$1^{2}$ .

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

Step 9.14.1.3

Reorder

$- 1^{2}$ and

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

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

Step 9.14.1.4

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 2 \cos (x)$ and

$b = 1$ .

$\sin (x) ((2 \cos (x) + 1) (2 \cos (x) - 1))$

Step 9.14.2

Remove unnecessary parentheses.

$\sin (x) (2 \cos (x) + 1) (2 \cos (x) - 1)$

Step 10

Apply the distributive property.

$(\sin (x) (2 \cos (x)) + \sin (x) \cdot 1) (2 \cos (x) - 1)$

Step 11

Simplify each term.

$(2 \sin (x) \cos (x) + \sin (x)) (2 \cos (x) - 1)$

Step 12

Apply the distributive property.

$(2 \sin (x) \cos (x) + \sin (x)) (2 \cos (x)) + (2 \sin (x) \cos (x) + \sin (x)) \cdot - 1$

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the distributive property.

$2 \sin (x) \cos (x) (2 \cos (x)) + \sin (x) (2 \cos (x)) + (2 \sin (x) \cos (x) + \sin (x)) \cdot - 1$

Step 13.1.2

Multiply

$2 \sin (x) \cos (x) (2 \cos (x))$ .

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Step 13.1.2.1

Multiply

$2$ by

$2$ .

$4 \sin (x) \cos (x) \cos (x) + \sin (x) (2 \cos (x)) + (2 \sin (x) \cos (x) + \sin (x)) \cdot - 1$

Step 13.1.2.2

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 13.1.2.3

Raise

$\cos (x)$ to the power of

$1$ .

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

Step 13.1.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 13.1.2.5

Add

$1$ and

$1$ .

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

Step 13.1.3

Move

$2$ to the left of

$\sin (x)$ .

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

Step 13.1.4

Apply the distributive property.

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

Step 13.1.5

Multiply

$- 1$ by

$2$ .

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

Step 13.1.6

Move

$- 1$ to the left of

$\sin (x)$ .

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

Step 13.1.7

Rewrite

$- 1 \sin (x)$ as

$- \sin (x)$ .

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

Step 13.2

Subtract

$2 \sin (x) \cos (x)$ from

$2 \sin (x) \cos (x)$ .

$4 \sin (x) {\cos (x)}^{2} + 0 - \sin (x)$

Step 13.3

Add

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

$0$ .

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

Step 14

Apply Pythagorean identity in reverse.

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

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Apply the distributive property.

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

Step 15.1.2

Multiply

$4$ by

$1$ .

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

Step 15.1.3

Multiply

$\sin (x)$ by

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

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Step 15.1.3.1

Move

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

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

Step 15.1.3.2

Multiply

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

$\sin (x)$ .

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Step 15.1.3.2.1

Raise

$\sin (x)$ to the power of

$1$ .

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

Step 15.1.3.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 15.1.3.3

Add

$2$ and

$1$ .

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

Step 15.1.4

Multiply

$- 1$ by

$4$ .

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

Step 15.2

Subtract

$\sin (x)$ from

$4 \sin (x)$ .

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

Step 16

Apply the sine triple-angle identity.

$\sin (3 x)$

Step 17

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (3 x) = 3 \sin (x) - 4 \sin^{3} (x)$ is an identity