Trigonometry Examples

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

Step 1

Simplify $\frac{3 + 3 \sin (x)}{\cos (x)} + \frac{3 \cos (x)}{1 + \sin (x)} - 12$ .

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Step 1.1

Factor $3$ out of $3 + 3 \sin (x)$ .

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Step 1.1.1

Factor $3$ out of $3$ .

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

Step 1.1.2

Factor $3$ out of $3 \cdot 1 + 3 \sin (x)$ .

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

Step 1.2

To write $\frac{3 (1 + \sin (x))}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Step 1.3

To write $\frac{3 \cos (x)}{1 + \sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 1.4

Write each expression with a common denominator of $\cos (x) (1 + \sin (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{3 (1 + \sin (x))}{\cos (x)}$ by $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Step 1.4.2

Multiply $\frac{3 \cos (x)}{1 + \sin (x)}$ by $\frac{\cos (x)}{\cos (x)}$ .

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

Step 1.4.3

Reorder the factors of $(1 + \sin (x)) \cos (x)$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $3$ out of $3 (1 + \sin (x)) (1 + \sin (x)) + 3 \cos (x) \cos (x)$ .

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Step 1.6.1.1.1

Factor $3$ out of $3 (1 + \sin (x)) (1 + \sin (x))$ .

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

Step 1.6.1.1.2

Factor $3$ out of $3 \cos (x) \cos (x)$ .

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

Step 1.6.1.1.3

Factor $3$ out of $3 ((1 + \sin (x)) (1 + \sin (x))) + 3 (\cos (x) \cos (x))$ .

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

Step 1.6.1.2

Expand $(1 + \sin (x)) (1 + \sin (x))$ using the FOIL Method.

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Step 1.6.1.2.1

Apply the distributive property.

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

Step 1.6.1.2.2

Apply the distributive property.

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

Step 1.6.1.2.3

Apply the distributive property.

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

Step 1.6.1.3

Simplify and combine like terms.

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Step 1.6.1.3.1

Simplify each term.

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Step 1.6.1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.6.1.3.1.2

Multiply $\sin (x)$ by $1$ .

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

Step 1.6.1.3.1.3

Multiply $\sin (x)$ by $1$ .

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

Step 1.6.1.3.1.4

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

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Step 1.6.1.3.1.4.1

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

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

Step 1.6.1.3.1.4.2

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

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

Step 1.6.1.3.1.4.3

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

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

Step 1.6.1.3.1.4.4

Add $1$ and $1$ .

$\frac{3 (1 + \sin (x) + \sin (x) + \sin^{2} (x) + \cos (x) \cos (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.3.2

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

$\frac{3 (1 + 2 \sin (x) + \sin^{2} (x) + \cos (x) \cos (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.4

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

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Step 1.6.1.4.1

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

$\frac{3 (1 + 2 \sin (x) + \sin^{2} (x) + \cos^{1} (x) \cos (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.4.2

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

$\frac{3 (1 + 2 \sin (x) + \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.4.3

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

$\frac{3 (1 + 2 \sin (x) + \sin^{2} (x) + {\cos (x)}^{1 + 1})}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.4.4

Add $1$ and $1$ .

$\frac{3 (1 + 2 \sin (x) + \sin^{2} (x) + \cos^{2} (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.5

Rewrite $1 + 2 \sin (x) + \sin^{2} (x) + \cos^{2} (x)$ in a factored form.

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Step 1.6.1.5.1

Apply pythagorean identity.

$\frac{3 (1 + 2 \sin (x) + 1)}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.5.2

Add $1$ and $1$ .

$\frac{3 (2 + 2 \sin (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.5.3

Factor $2$ out of $2 + 2 \sin (x)$ .

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Step 1.6.1.5.3.1

Factor $2$ out of $2$ .

$\frac{3 (2 \cdot 1 + 2 \sin (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.5.3.2

Factor $2$ out of $2 \cdot 1 + 2 \sin (x)$ .

$\frac{3 (2 (1 + \sin (x)))}{\cos (x) (1 + \sin (x))} - 12 = 0$

$\frac{3 \cdot 2 (1 + \sin (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.1.6

Multiply $3$ by $2$ .

$\frac{6 (1 + \sin (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.2

Cancel the common factor of $1 + \sin (x)$ .

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Step 1.6.2.1

Cancel the common factor.

$\frac{6 (1 + \sin (x))}{\cos (x) (1 + \sin (x))} - 12 = 0$

Step 1.6.2.2

Rewrite the expression.

$\frac{6}{\cos (x)} - 12 = 0$

Step 1.6.3

Separate fractions.

$\frac{6}{1} \cdot \frac{1}{\cos (x)} - 12 = 0$

Step 1.6.4

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

$\frac{6}{1} \sec (x) - 12 = 0$

Step 1.6.5

Divide $6$ by $1$ .

$6 \sec (x) - 12 = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 3