Trigonometry Examples

\sin (2 x)

$\sin (2 x)$

Step 1

A good method to expand

\sin (2 x)

$\sin (2 x)$ is by using De Moivre's theorem

(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))

$(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . When

r = 1

$r = 1$ ,

\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$

Step 2

Expand the right hand side of

\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}

$\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ using the binomial theorem.

Expand:

{(\cos (x) + i \cdot \sin (x))}^{2}

${(\cos (x) + i \cdot \sin (x))}^{2}$

Step 3

Rewrite

{(\cos (x) + i \sin (x))}^{2}

${(\cos (x) + i \sin (x))}^{2}$ as

(\cos (x) + i \sin (x)) (\cos (x) + i \sin (x))

$(\cos (x) + i \sin (x)) (\cos (x) + i \sin (x))$ .

(\cos (x) + i \sin (x)) (\cos (x) + i \sin (x))

$(\cos (x) + i \sin (x)) (\cos (x) + i \sin (x))$

Step 4

Expand

(\cos (x) + i \sin (x)) (\cos (x) + i \sin (x))

$(\cos (x) + i \sin (x)) (\cos (x) + i \sin (x))$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

\cos (x) (\cos (x) + i \sin (x)) + i \sin (x) (\cos (x) + i \sin (x))

$\cos (x) (\cos (x) + i \sin (x)) + i \sin (x) (\cos (x) + i \sin (x))$

Step 4.2

Apply the distributive property.

\cos (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) (\cos (x) + i \sin (x))

$\cos (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) (\cos (x) + i \sin (x))$

Step 4.3

Apply the distributive property.

\cos (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

$\cos (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))$

\cos (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

$\cos (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply

\cos (x) \cos (x)

$\cos (x) \cos (x)$ .

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Step 5.1.1.1

Raise

\cos (x)

$\cos (x)$ to the power of

1

$1$ .

\cos^{1} (x) \cos (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

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

Step 5.1.1.2

Raise

\cos (x)

$\cos (x)$ to the power of

1

$1$ .

\cos^{1} (x) \cos^{1} (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

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

Step 5.1.1.3

Use the power rule

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

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

{\cos (x)}^{1 + 1} + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

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

Step 5.1.1.4

Add

1

$1$ and

1

$1$ .

\cos^{2} (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

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

\cos^{2} (x) + \cos (x) (i \sin (x)) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

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

Step 5.1.2

Rewrite using the commutative property of multiplication.

\cos^{2} (x) + i \cos (x) \sin (x) + i \sin (x) \cos (x) + i \sin (x) (i \sin (x))

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

Step 5.1.3

Multiply

i \sin (x) (i \sin (x))

$i \sin (x) (i \sin (x))$ .

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Step 5.1.3.1

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 5.1.3.2

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 5.1.3.3

Use the power rule

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

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

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

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

Step 5.1.3.4

Add

1

$1$ and

1

$1$ .

\cos^{2} (x) + i \cos (x) \sin (x) + i \sin (x) \cos (x) + i^{2} \sin (x) \sin (x)

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

Step 5.1.3.5

Raise

\sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 5.1.3.6

Raise

\sin (x)

$\sin (x)$ to the power of

1

$1$ .

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

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

Step 5.1.3.7

Use the power rule

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

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

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

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

Step 5.1.3.8

Add

1

$1$ and

1

$1$ .

\cos^{2} (x) + i \cos (x) \sin (x) + i \sin (x) \cos (x) + i^{2} \sin^{2} (x)

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

\cos^{2} (x) + i \cos (x) \sin (x) + i \sin (x) \cos (x) + i^{2} \sin^{2} (x)

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

Step 5.1.4

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 5.1.5

Rewrite

- 1 \sin^{2} (x)

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

- \sin^{2} (x)

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

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

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

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

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

Step 5.2

Reorder the factors of

i \sin (x) \cos (x)

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

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

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

Step 5.3

Add

i \cos (x) \sin (x)

$i \cos (x) \sin (x)$ and

i \cos (x) \sin (x)

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

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

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

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

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

Step 6

Move

- \sin^{2} (x)

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

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

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

Step 7

Apply the cosine double-angle identity.

\cos (2 x) + 2 i \cos (x) \sin (x)

$\cos (2 x) + 2 i \cos (x) \sin (x)$

Step 8

Simplify each term.

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Step 8.1

Add parentheses.

\cos (2 x) + 2 i (\cos (x) \sin (x))

$\cos (2 x) + 2 i (\cos (x) \sin (x))$

Step 8.2

Reorder

2 i

$2 i$ and

\cos (x) \sin (x)

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

\cos (2 x) + \cos (x) \sin (x) (2 i)

$\cos (2 x) + \cos (x) \sin (x) (2 i)$

Step 8.3

Add parentheses.

\cos (2 x) + \cos (x) (\sin (x) \cdot 2) i

$\cos (2 x) + \cos (x) (\sin (x) \cdot 2) i$

Step 8.4

Reorder

\cos (x)

$\cos (x)$ and

\sin (x) \cdot 2

$\sin (x) \cdot 2$ .

\cos (2 x) + \sin (x) \cdot 2 \cos (x) i

$\cos (2 x) + \sin (x) \cdot 2 \cos (x) i$

Step 8.5

Reorder

\sin (x)

$\sin (x)$ and

2

$2$ .

\cos (2 x) + 2 \cdot \sin (x) \cos (x) i

$\cos (2 x) + 2 \cdot \sin (x) \cos (x) i$

Step 8.6

Apply the sine double-angle identity.

\cos (2 x) + \sin (2 x) i

$\cos (2 x) + \sin (2 x) i$

\cos (2 x) + \sin (2 x) i

$\cos (2 x) + \sin (2 x) i$

Step 9

Reorder factors in

\cos (2 x) + \sin (2 x) i

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

\cos (2 x) + i \sin (2 x)

$\cos (2 x) + i \sin (2 x)$

Step 10

Take out the expressions with the imaginary part, which are equal to

\sin (2 x)

$\sin (2 x)$ . Remove the imaginary number

i

$i$ .

\sin (2 x) = 2 \sin (x) \cos (x)

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