Trigonometry Examples

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

Step 1

Rewrite ${(3 \sin (x) - 3 \cos (x))}^{2}$ as $(3 \sin (x) - 3 \cos (x)) (3 \sin (x) - 3 \cos (x))$ .

$(3 \sin (x) - 3 \cos (x)) (3 \sin (x) - 3 \cos (x))$

Step 2

Expand $(3 \sin (x) - 3 \cos (x)) (3 \sin (x) - 3 \cos (x))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 \sin (x) (3 \sin (x) - 3 \cos (x)) - 3 \cos (x) (3 \sin (x) - 3 \cos (x))$

Step 2.2

Apply the distributive property.

$3 \sin (x) (3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x) - 3 \cos (x))$

Step 2.3

Apply the distributive property.

$3 \sin (x) (3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $3 \sin (x) (3 \sin (x))$ .

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Step 3.1.1.1

Multiply $3$ by $3$ .

$9 \sin (x) \sin (x) + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.1.2

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

$9 (\sin^{1} (x) \sin (x)) + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.1.3

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

$9 (\sin^{1} (x) \sin^{1} (x)) + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.1.4

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

$9 {\sin (x)}^{1 + 1} + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.1.5

Add $1$ and $1$ .

$9 \sin^{2} (x) + 3 \sin (x) (- 3 \cos (x)) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.2

Multiply $- 3$ by $3$ .

$9 \sin^{2} (x) - 9 \sin (x) \cos (x) - 3 \cos (x) (3 \sin (x)) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.3

Multiply $3$ by $- 3$ .

$9 \sin^{2} (x) - 9 \sin (x) \cos (x) - 9 \cos (x) \sin (x) - 3 \cos (x) (- 3 \cos (x))$

Step 3.1.4

Multiply $- 3 \cos (x) (- 3 \cos (x))$ .

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Step 3.1.4.1

Multiply $- 3$ by $- 3$ .

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

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

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

Step 3.1.4.5

Add $1$ and $1$ .

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

Step 3.2

Reorder the factors of $- 9 \sin (x) \cos (x)$ .

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

Step 3.3

Subtract $9 \cos (x) \sin (x)$ from $- 9 \cos (x) \sin (x)$ .

$9 \sin^{2} (x) - 18 \cos (x) \sin (x) + 9 \cos^{2} (x)$

Step 4

Simplify with factoring out.

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Step 4.1

Move $9 \cos^{2} (x)$ .

$9 \sin^{2} (x) + 9 \cos^{2} (x) - 18 \cos (x) \sin (x)$

Step 4.2

Factor $9$ out of $9 \sin^{2} (x)$ .

$9 (\sin^{2} (x)) + 9 \cos^{2} (x) - 18 \cos (x) \sin (x)$

Step 4.3

Factor $9$ out of $9 \cos^{2} (x)$ .

$9 (\sin^{2} (x)) + 9 (\cos^{2} (x)) - 18 \cos (x) \sin (x)$

Step 4.4

Factor $9$ out of $9 (\sin^{2} (x)) + 9 (\cos^{2} (x))$ .

$9 (\sin^{2} (x) + \cos^{2} (x)) - 18 \cos (x) \sin (x)$

Step 5

Apply pythagorean identity.

$9 \cdot 1 - 18 \cos (x) \sin (x)$

Step 6

Multiply $9$ by $1$ .

$9 - 18 \cos (x) \sin (x)$