Trigonometry Examples

$\sin (2 θ)$

Step 1

A good method to expand $\sin (2 θ)$ is by using De Moivre's theorem $(r {(\cos (x) + i \cdot \sin (x))}^{n} = r^{n} (\cos (n x) + i \cdot \sin (n x)))$ . When $r = 1$ , $\cos (n x) + i \cdot \sin (n x) = {(\cos (x) + i \cdot \sin (x))}^{n}$ .

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

Step 2

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

Expand: ${(\cos (θ) + i \cdot \sin (θ))}^{2}$

Step 3

Rewrite ${(\cos (θ) + i \sin (θ))}^{2}$ as $(\cos (θ) + i \sin (θ)) (\cos (θ) + i \sin (θ))$ .

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

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 $\cos (θ) \cos (θ)$ .

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Step 5.1.1.1

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

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

Step 5.1.1.2

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

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

Step 5.1.1.3

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

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

Step 5.1.1.4

Add $1$ and $1$ .

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

Step 5.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.3

Multiply $i \sin (θ) (i \sin (θ))$ .

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Step 5.1.3.1

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

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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

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

Step 5.1.3.4

Add $1$ and $1$ .

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

Step 5.1.3.5

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

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

Step 5.1.3.6

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

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

Step 5.1.3.7

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

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

Step 5.1.3.8

Add $1$ and $1$ .

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

Step 5.1.4

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

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

Step 5.1.5

Rewrite $- 1 \sin^{2} (θ)$ as $- \sin^{2} (θ)$ .

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

Step 5.2

Reorder the factors of $i \sin (θ) \cos (θ)$ .

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

Step 5.3

Add $i \cos (θ) \sin (θ)$ and $i \cos (θ) \sin (θ)$ .

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

Step 6

Move $- \sin^{2} (θ)$ .

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

Step 7

Apply the cosine double-angle identity.

$\cos (2 θ) + 2 i \cos (θ) \sin (θ)$

Step 8

Simplify each term.

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Step 8.1

Add parentheses.

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

Step 8.2

Reorder $2 i$ and $\cos (θ) \sin (θ)$ .

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

Step 8.3

Add parentheses.

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

Step 8.4

Reorder $\cos (θ)$ and $\sin (θ) \cdot 2$ .

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

Step 8.5

Reorder $\sin (θ)$ and $2$ .

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

Step 8.6

Apply the sine double-angle identity.

$\cos (2 θ) + \sin (2 θ) i$

Step 9

Reorder factors in $\cos (2 θ) + \sin (2 θ) i$ .

$\cos (2 θ) + i \sin (2 θ)$

Step 10

Take out the expressions with the imaginary part, which are equal to $\sin (2 θ)$ . Remove the imaginary number $i$ .

$\sin (2 θ) = 2 \sin (θ) \cos (θ)$