Trigonometry Examples

$\cos^{2} (\frac{θ}{3}) - \sin^{2} (\frac{θ}{3})$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (\frac{θ}{3})$ and $b = \sin (\frac{θ}{3})$ .

$(\cos (\frac{θ}{3}) + \sin (\frac{θ}{3})) (\cos (\frac{θ}{3}) - \sin (\frac{θ}{3}))$

Step 2

Expand $(\cos (\frac{θ}{3}) + \sin (\frac{θ}{3})) (\cos (\frac{θ}{3}) - \sin (\frac{θ}{3}))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\cos (\frac{θ}{3}) (\cos (\frac{θ}{3}) - \sin (\frac{θ}{3})) + \sin (\frac{θ}{3}) (\cos (\frac{θ}{3}) - \sin (\frac{θ}{3}))$

Step 2.2

Apply the distributive property.

$\cos (\frac{θ}{3}) \cos (\frac{θ}{3}) + \cos (\frac{θ}{3}) (- \sin (\frac{θ}{3})) + \sin (\frac{θ}{3}) (\cos (\frac{θ}{3}) - \sin (\frac{θ}{3}))$

Step 2.3

Apply the distributive property.

$\cos (\frac{θ}{3}) \cos (\frac{θ}{3}) + \cos (\frac{θ}{3}) (- \sin (\frac{θ}{3})) + \sin (\frac{θ}{3}) \cos (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $\cos (\frac{θ}{3}) \cos (\frac{θ}{3}) + \cos (\frac{θ}{3}) (- \sin (\frac{θ}{3})) + \sin (\frac{θ}{3}) \cos (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$ .

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Step 3.1.1

Reorder the factors in the terms $\cos (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$ and $\sin (\frac{θ}{3}) \cos (\frac{θ}{3})$ .

$\cos (\frac{θ}{3}) \cos (\frac{θ}{3}) - \cos (\frac{θ}{3}) \sin (\frac{θ}{3}) + \cos (\frac{θ}{3}) \sin (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.1.2

Add $- \cos (\frac{θ}{3}) \sin (\frac{θ}{3})$ and $\cos (\frac{θ}{3}) \sin (\frac{θ}{3})$ .

$\cos (\frac{θ}{3}) \cos (\frac{θ}{3}) + 0 + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.1.3

Add $\cos (\frac{θ}{3}) \cos (\frac{θ}{3})$ and $0$ .

$\cos (\frac{θ}{3}) \cos (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $\cos (\frac{θ}{3}) \cos (\frac{θ}{3})$ .

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Step 3.2.1.1

Raise $\cos (\frac{θ}{3})$ to the power of $1$ .

$\cos^{1} (\frac{θ}{3}) \cos (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.2.1.2

Raise $\cos (\frac{θ}{3})$ to the power of $1$ .

$\cos^{1} (\frac{θ}{3}) \cos^{1} (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.2.1.3

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

${\cos (\frac{θ}{3})}^{1 + 1} + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.2.1.4

Add $1$ and $1$ .

$\cos^{2} (\frac{θ}{3}) + \sin (\frac{θ}{3}) (- \sin (\frac{θ}{3}))$

Step 3.2.2

Rewrite using the commutative property of multiplication.

$\cos^{2} (\frac{θ}{3}) - \sin (\frac{θ}{3}) \sin (\frac{θ}{3})$

Step 3.2.3

Multiply $- \sin (\frac{θ}{3}) \sin (\frac{θ}{3})$ .

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Step 3.2.3.1

Raise $\sin (\frac{θ}{3})$ to the power of $1$ .

$\cos^{2} (\frac{θ}{3}) - (\sin^{1} (\frac{θ}{3}) \sin (\frac{θ}{3}))$

Step 3.2.3.2

Raise $\sin (\frac{θ}{3})$ to the power of $1$ .

$\cos^{2} (\frac{θ}{3}) - (\sin^{1} (\frac{θ}{3}) \sin^{1} (\frac{θ}{3}))$

Step 3.2.3.3

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

$\cos^{2} (\frac{θ}{3}) - {\sin (\frac{θ}{3})}^{1 + 1}$

Step 3.2.3.4

Add $1$ and $1$ .

$\cos^{2} (\frac{θ}{3}) - \sin^{2} (\frac{θ}{3})$

Step 4

Apply the cosine double-angle identity.

$\cos (2 \frac{θ}{3})$

Step 5

Combine $2$ and $\frac{θ}{3}$ .

$\cos (\frac{2 θ}{3})$