Algebra Examples

${(- 2 \cos (x) - 3 \sin (x))}^{2} - 5 \sin^{2} (x) = 1$

Step 1

Subtract $1$ from both sides of the equation.

${(- 2 \cos (x) - 3 \sin (x))}^{2} - 5 \sin^{2} (x) - 1 = 0$

Step 2

Simplify ${(- 2 \cos (x) - 3 \sin (x))}^{2} - 5 \sin^{2} (x) - 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(- 2 \cos (x) - 3 \sin (x)) (- 2 \cos (x) - 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.2

Expand $(- 2 \cos (x) - 3 \sin (x)) (- 2 \cos (x) - 3 \sin (x))$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$- 2 \cos (x) (- 2 \cos (x) - 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x) - 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.2.2

Apply the distributive property.

$- 2 \cos (x) (- 2 \cos (x)) - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x) - 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.2.3

Apply the distributive property.

$- 2 \cos (x) (- 2 \cos (x)) - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $- 2 \cos (x) (- 2 \cos (x))$ .

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Step 2.1.3.1.1.1

Multiply $- 2$ by $- 2$ .

$4 \cos (x) \cos (x) - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.1.2

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

$4 (\cos^{1} (x) \cos (x)) - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.1.3

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

$4 (\cos^{1} (x) \cos^{1} (x)) - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.1.4

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

$4 {\cos (x)}^{1 + 1} - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.1.5

Add $1$ and $1$ .

$4 \cos^{2} (x) - 2 \cos (x) (- 3 \sin (x)) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.2

Multiply $- 3$ by $- 2$ .

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) - 3 \sin (x) (- 2 \cos (x)) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.3

Multiply $- 2$ by $- 3$ .

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \sin (x) \cos (x) - 3 \sin (x) (- 3 \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.4

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

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Step 2.1.3.1.4.1

Multiply $- 3$ by $- 3$ .

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \sin (x) \cos (x) + 9 \sin (x) \sin (x) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.4.2

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

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \sin (x) \cos (x) + 9 (\sin^{1} (x) \sin (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.4.3

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

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \sin (x) \cos (x) + 9 (\sin^{1} (x) \sin^{1} (x)) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.4.4

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

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \sin (x) \cos (x) + 9 {\sin (x)}^{1 + 1} - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.1.4.5

Add $1$ and $1$ .

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \sin (x) \cos (x) + 9 \sin^{2} (x) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.2

Reorder the factors of $6 \sin (x) \cos (x)$ .

$4 \cos^{2} (x) + 6 \cos (x) \sin (x) + 6 \cos (x) \sin (x) + 9 \sin^{2} (x) - 5 \sin^{2} (x) - 1 = 0$

Step 2.1.3.3

Add $6 \cos (x) \sin (x)$ and $6 \cos (x) \sin (x)$ .

$4 \cos^{2} (x) + 12 \cos (x) \sin (x) + 9 \sin^{2} (x) - 5 \sin^{2} (x) - 1 = 0$

Step 2.2

Subtract $5 \sin^{2} (x)$ from $9 \sin^{2} (x)$ .

$4 \cos^{2} (x) + 12 \cos (x) \sin (x) + 4 \sin^{2} (x) - 1 = 0$

Step 2.3

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

$4 \cos^{2} (x) + 4 \sin^{2} (x) + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.4

Factor $4$ out of $4 \cos^{2} (x)$ .

$4 (\cos^{2} (x)) + 4 \sin^{2} (x) + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.5

Factor $4$ out of $4 \sin^{2} (x)$ .

$4 (\cos^{2} (x)) + 4 (\sin^{2} (x)) + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.6

Factor $4$ out of $4 (\cos^{2} (x)) + 4 (\sin^{2} (x))$ .

$4 (\cos^{2} (x) + \sin^{2} (x)) + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.7

Rearrange terms.

$4 (\sin^{2} (x) + \cos^{2} (x)) + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.8

Apply pythagorean identity.

$4 \cdot 1 + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.9

Multiply $4$ by $1$ .

$4 + 12 \cos (x) \sin (x) - 1 = 0$

Step 2.10

Subtract $1$ from $4$ .

$12 \cos (x) \sin (x) + 3 = 0$

Step 3

Divide each term in the equation by $\cos (x)$ .

$\frac{12 \cos (x) \sin (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4

Cancel the common factor of $\cos (x)$ .

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Step 4.1

Cancel the common factor.

$\frac{12 \cos (x) \sin (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4.2

Divide $12 \sin (x)$ by $1$ .

$12 \sin (x) + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5

Separate fractions.

$12 \sin (x) + \frac{3}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + \frac{3}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 7

Divide $3$ by $1$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{\cos (x)}$

Step 8

Separate fractions.

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 9

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 10

Divide $0$ by $1$ .

$12 \sin (x) + 3 \sec (x) = 0 \sec (x)$

Step 11

Multiply $0$ by $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = 0$

Step 12

Multiply both sides by $\cos (x)$ .

$(12 \sin (x) + 3 \sec (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 13

Simplify.

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Step 13.1

Simplify the left side.

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Step 13.1.1

Simplify $(12 \sin (x) + 3 \sec (x)) \cos (x)$ .

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Step 13.1.1.1

Simplify each term.

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Step 13.1.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$(12 \sin (x) + 3 \frac{1}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 13.1.1.1.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

$(12 \sin (x) + \frac{3}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 13.1.1.2

Simplify terms.

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Step 13.1.1.2.1

Apply the distributive property.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 13.1.1.2.2

Cancel the common factor of $\cos (x)$ .

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Step 13.1.1.2.2.1

Cancel the common factor.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 13.1.1.2.2.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 13.2

Simplify the right side.

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Step 13.2.1

Cancel the common factor of $\cos (x)$ .

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Step 13.2.1.1

Cancel the common factor.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 13.2.1.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = 0$

Step 14

Solve for $x$ .

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Step 14.1

Divide each term in the equation by $\cos (x)$ .

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.2

Cancel the common factor of $\cos (x)$ .

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Step 14.2.1

Cancel the common factor.

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.2.2

Divide $12 \sin (x)$ by $1$ .

$12 \sin (x) + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.3

Separate fractions.

$12 \sin (x) + \frac{3}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + \frac{3}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 14.5

Divide $3$ by $1$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{\cos (x)}$

Step 14.6

Separate fractions.

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 14.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 14.8

Divide $0$ by $1$ .

$12 \sin (x) + 3 \sec (x) = 0 \sec (x)$

Step 14.9

Multiply $0$ by $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = 0$

Step 14.10

Multiply both sides by $\cos (x)$ .

$(12 \sin (x) + 3 \sec (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.11

Simplify.

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Step 14.11.1

Simplify the left side.

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Step 14.11.1.1

Simplify $(12 \sin (x) + 3 \sec (x)) \cos (x)$ .

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Step 14.11.1.1.1

Simplify each term.

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Step 14.11.1.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$(12 \sin (x) + 3 \frac{1}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.11.1.1.1.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

$(12 \sin (x) + \frac{3}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.11.1.1.2

Simplify terms.

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Step 14.11.1.1.2.1

Apply the distributive property.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.11.1.1.2.2

Cancel the common factor of $\cos (x)$ .

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Step 14.11.1.1.2.2.1

Cancel the common factor.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.11.1.1.2.2.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.11.2

Simplify the right side.

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Step 14.11.2.1

Cancel the common factor of $\cos (x)$ .

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Step 14.11.2.1.1

Cancel the common factor.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.11.2.1.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = 0$

Step 14.12

Solve for $x$ .

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Step 14.12.1

Divide each term in the equation by $\cos (x)$ .

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.2

Cancel the common factor of $\cos (x)$ .

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Step 14.12.2.1

Cancel the common factor.

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.2.2

Divide $12 \sin (x)$ by $1$ .

$12 \sin (x) + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.3

Separate fractions.

$12 \sin (x) + \frac{3}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + \frac{3}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 14.12.5

Divide $3$ by $1$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{\cos (x)}$

Step 14.12.6

Separate fractions.

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 14.12.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 14.12.8

Divide $0$ by $1$ .

$12 \sin (x) + 3 \sec (x) = 0 \sec (x)$

Step 14.12.9

Multiply $0$ by $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = 0$

Step 14.12.10

Multiply both sides by $\cos (x)$ .

$(12 \sin (x) + 3 \sec (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11

Simplify.

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Step 14.12.11.1

Simplify the left side.

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Step 14.12.11.1.1

Simplify $(12 \sin (x) + 3 \sec (x)) \cos (x)$ .

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Step 14.12.11.1.1.1

Simplify each term.

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Step 14.12.11.1.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$(12 \sin (x) + 3 \frac{1}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11.1.1.1.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

$(12 \sin (x) + \frac{3}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11.1.1.2

Simplify terms.

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Step 14.12.11.1.1.2.1

Apply the distributive property.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11.1.1.2.2

Cancel the common factor of $\cos (x)$ .

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Step 14.12.11.1.1.2.2.1

Cancel the common factor.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11.1.1.2.2.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11.2

Simplify the right side.

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Step 14.12.11.2.1

Cancel the common factor of $\cos (x)$ .

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Step 14.12.11.2.1.1

Cancel the common factor.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.11.2.1.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = 0$

Step 14.12.12

Solve for $x$ .

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Step 14.12.12.1

Divide each term in the equation by $\cos (x)$ .

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.2

Cancel the common factor of $\cos (x)$ .

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Step 14.12.12.2.1

Cancel the common factor.

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.2.2

Divide $12 \sin (x)$ by $1$ .

$12 \sin (x) + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.3

Separate fractions.

$12 \sin (x) + \frac{3}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + \frac{3}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 14.12.12.5

Divide $3$ by $1$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{\cos (x)}$

Step 14.12.12.6

Separate fractions.

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 14.12.12.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 14.12.12.8

Divide $0$ by $1$ .

$12 \sin (x) + 3 \sec (x) = 0 \sec (x)$

Step 14.12.12.9

Multiply $0$ by $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = 0$

Step 14.12.12.10

Multiply both sides by $\cos (x)$ .

$(12 \sin (x) + 3 \sec (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11

Simplify.

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Step 14.12.12.11.1

Simplify the left side.

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Step 14.12.12.11.1.1

Simplify $(12 \sin (x) + 3 \sec (x)) \cos (x)$ .

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Step 14.12.12.11.1.1.1

Simplify each term.

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Step 14.12.12.11.1.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$(12 \sin (x) + 3 \frac{1}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11.1.1.1.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

$(12 \sin (x) + \frac{3}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11.1.1.2

Simplify terms.

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Step 14.12.12.11.1.1.2.1

Apply the distributive property.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11.1.1.2.2

Cancel the common factor of $\cos (x)$ .

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Step 14.12.12.11.1.1.2.2.1

Cancel the common factor.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11.1.1.2.2.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11.2

Simplify the right side.

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Step 14.12.12.11.2.1

Cancel the common factor of $\cos (x)$ .

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Step 14.12.12.11.2.1.1

Cancel the common factor.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.11.2.1.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = 0$

Step 14.12.12.12

Solve for $x$ .

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Step 14.12.12.12.1

Divide each term in the equation by $\cos (x)$ .

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.12.2

Cancel the common factor of $\cos (x)$ .

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Step 14.12.12.12.2.1

Cancel the common factor.

$\frac{12 \sin (x) \cos (x)}{\cos (x)} + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.12.2.2

Divide $12 \sin (x)$ by $1$ .

$12 \sin (x) + \frac{3}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.12.3

Separate fractions.

$12 \sin (x) + \frac{3}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.12.12.12.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + \frac{3}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 14.12.12.12.5

Divide $3$ by $1$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{\cos (x)}$

Step 14.12.12.12.6

Separate fractions.

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 14.12.12.12.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 14.12.12.12.8

Divide $0$ by $1$ .

$12 \sin (x) + 3 \sec (x) = 0 \sec (x)$

Step 14.12.12.12.9

Multiply $0$ by $\sec (x)$ .

$12 \sin (x) + 3 \sec (x) = 0$

Step 14.12.12.12.10

Multiply both sides by $\cos (x)$ .

$(12 \sin (x) + 3 \sec (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11

Simplify.

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Step 14.12.12.12.11.1

Simplify the left side.

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Step 14.12.12.12.11.1.1

Simplify $(12 \sin (x) + 3 \sec (x)) \cos (x)$ .

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Step 14.12.12.12.11.1.1.1

Simplify each term.

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Step 14.12.12.12.11.1.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$(12 \sin (x) + 3 \frac{1}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11.1.1.1.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

$(12 \sin (x) + \frac{3}{\cos (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11.1.1.2

Simplify terms.

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Step 14.12.12.12.11.1.1.2.1

Apply the distributive property.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11.1.1.2.2

Cancel the common factor of $\cos (x)$ .

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Step 14.12.12.12.11.1.1.2.2.1

Cancel the common factor.

$12 \sin (x) \cos (x) + \frac{3}{\cos (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11.1.1.2.2.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11.2

Simplify the right side.

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Step 14.12.12.12.11.2.1

Cancel the common factor of $\cos (x)$ .

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Step 14.12.12.12.11.2.1.1

Cancel the common factor.

$12 \sin (x) \cos (x) + 3 = \frac{0}{\cos (x)} \cos (x)$

Step 14.12.12.12.11.2.1.2

Rewrite the expression.

$12 \sin (x) \cos (x) + 3 = 0$

Step 14.12.12.12.12

Solve for $x$ .

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Step 14.12.12.12.12.1

Subtract $3$ from both sides of the equation.

$12 \sin (x) \cos (x) = - 3$

Step 14.12.12.12.12.2

Multiply each term in $12 \sin (x) \cos (x) = - 3$ by $\frac{1}{6}$ to eliminate the fractions.

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Step 14.12.12.12.12.2.1

Multiply each term in $12 \sin (x) \cos (x) = - 3$ by $\frac{1}{6}$ .

$12 \sin (x) \cos (x) \frac{1}{6} = - 3 (\frac{1}{6})$

Step 14.12.12.12.12.2.2

Simplify the left side.

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Step 14.12.12.12.12.2.2.1

Cancel the common factor of $6$ .

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Step 14.12.12.12.12.2.2.1.1

Factor $6$ out of $12 \sin (x) \cos (x)$ .

$6 (2 \sin (x) \cos (x)) \frac{1}{6} = - 3 (\frac{1}{6})$

Step 14.12.12.12.12.2.2.1.2

Cancel the common factor.

$6 (2 \sin (x) \cos (x)) \frac{1}{6} = - 3 (\frac{1}{6})$

Step 14.12.12.12.12.2.2.1.3

Rewrite the expression.

$2 \sin (x) \cos (x) = - 3 (\frac{1}{6})$

Step 14.12.12.12.12.2.2.2

Apply the sine double-angle identity.

$\sin (2 x) = - 3 (\frac{1}{6})$

Step 14.12.12.12.12.2.3

Simplify the right side.

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Step 14.12.12.12.12.2.3.1

Cancel the common factor of $3$ .

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Step 14.12.12.12.12.2.3.1.1

Factor $3$ out of $- 3$ .

$\sin (2 x) = 3 (- 1) \frac{1}{6}$

Step 14.12.12.12.12.2.3.1.2

Factor $3$ out of $6$ .

$\sin (2 x) = 3 \cdot - 1 \frac{1}{3 \cdot 2}$

Step 14.12.12.12.12.2.3.1.3

Cancel the common factor.

$\sin (2 x) = 3 \cdot - 1 \frac{1}{3 \cdot 2}$

Step 14.12.12.12.12.2.3.1.4

Rewrite the expression.

$\sin (2 x) = - \frac{1}{2}$

Step 14.12.12.12.12.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$2 x = \arcsin (- \frac{1}{2})$

Step 14.12.12.12.12.4

Simplify the right side.

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Step 14.12.12.12.12.4.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$2 x = - \frac{π}{6}$

Step 14.12.12.12.12.5

Divide each term in $2 x = - \frac{π}{6}$ by $2$ and simplify.

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Step 14.12.12.12.12.5.1

Divide each term in $2 x = - \frac{π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{- \frac{π}{6}}{2}$

Step 14.12.12.12.12.5.2

Simplify the left side.

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Step 14.12.12.12.12.5.2.1

Cancel the common factor of $2$ .

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Step 14.12.12.12.12.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- \frac{π}{6}}{2}$

Step 14.12.12.12.12.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{6}}{2}$

Step 14.12.12.12.12.5.3

Simplify the right side.

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Step 14.12.12.12.12.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{π}{6} \cdot \frac{1}{2}$

Step 14.12.12.12.12.5.3.2

Multiply $- \frac{π}{6} \cdot \frac{1}{2}$ .

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Step 14.12.12.12.12.5.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{6}$ .

$x = - \frac{π}{2 \cdot 6}$

Step 14.12.12.12.12.5.3.2.2

Multiply $2$ by $6$ .

$x = - \frac{π}{12}$

Step 14.12.12.12.12.6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$2 x = 2 π + \frac{π}{6} + π$

Step 14.12.12.12.12.7

Simplify the expression to find the second solution.

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Step 14.12.12.12.12.7.1

Subtract $2 π$ from $2 π + \frac{π}{6} + π$ .

$2 x = 2 π + \frac{π}{6} + π - 2 π$

Step 14.12.12.12.12.7.2

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{6} + π$ .

$2 x = \frac{7 π}{6}$

Step 14.12.12.12.12.7.3

Divide each term in $2 x = \frac{7 π}{6}$ by $2$ and simplify.

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Step 14.12.12.12.12.7.3.1

Divide each term in $2 x = \frac{7 π}{6}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2}$

Step 14.12.12.12.12.7.3.2

Simplify the left side.

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Step 14.12.12.12.12.7.3.2.1

Cancel the common factor of $2$ .

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Step 14.12.12.12.12.7.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2}$

Step 14.12.12.12.12.7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{2}$

Step 14.12.12.12.12.7.3.3

Simplify the right side.

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Step 14.12.12.12.12.7.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{2}$

Step 14.12.12.12.12.7.3.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{2}$ .

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Step 14.12.12.12.12.7.3.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{2}$ .

$x = \frac{7 π}{6 \cdot 2}$

Step 14.12.12.12.12.7.3.3.2.2

Multiply $6$ by $2$ .

$x = \frac{7 π}{12}$

Step 14.12.12.12.12.8

Find the period of $\sin (2 x)$ .

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Step 14.12.12.12.12.8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 14.12.12.12.12.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 14.12.12.12.12.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 14.12.12.12.12.8.4

Cancel the common factor of $2$ .

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Step 14.12.12.12.12.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 14.12.12.12.12.8.4.2

Divide $π$ by $1$ .

$π$

Step 14.12.12.12.12.9

Add $π$ to every negative angle to get positive angles.

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Step 14.12.12.12.12.9.1

Add $π$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + π$

Step 14.12.12.12.12.9.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$π \cdot \frac{12}{12} - \frac{π}{12}$

Step 14.12.12.12.12.9.3

Combine fractions.

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Step 14.12.12.12.12.9.3.1

Combine $π$ and $\frac{12}{12}$ .

$\frac{π \cdot 12}{12} - \frac{π}{12}$

Step 14.12.12.12.12.9.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 12 - π}{12}$

Step 14.12.12.12.12.9.4

Simplify the numerator.

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Step 14.12.12.12.12.9.4.1

Move $12$ to the left of $π$ .

$\frac{12 \cdot π - π}{12}$

Step 14.12.12.12.12.9.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{12}$

Step 14.12.12.12.12.9.5

List the new angles.

$x = \frac{11 π}{12}$

Step 14.12.12.12.12.10

The period of the $\sin (2 x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{7 π}{12} + π n, \frac{11 π}{12} + π n$ , for any integer $n$