Precalculus Examples

$(\tan^{2} (x) - 9) (2 \cos (x) + 1) = 0$

Step 1

Simplify $(\tan^{2} (x) - 9) (2 \cos (x) + 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.2

Expand $(\frac{\sin^{2} (x)}{\cos^{2} (x)} - 9) (2 \cos (x) + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} (2 \cos (x)) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x) + 1) = 0$

Step 1.2.3

Apply the distributive property.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} (2 \cos (x)) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{\sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3.1.2

Combine $2$ and $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$\frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3.1.3

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

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Step 1.3.1.3.1

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

$\frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3.1.3.2

Cancel the common factor.

$\frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3.1.3.3

Rewrite the expression.

$\frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot 1 - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3.1.4

Multiply $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ by $1$ .

$\frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 9 (2 \cos (x)) - 9 \cdot 1 = 0$

Step 1.3.1.5

Multiply $2$ by $- 9$ .

$\frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 18 \cos (x) - 9 \cdot 1 = 0$

Step 1.3.1.6

Multiply $- 9$ by $1$ .

$\frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 18 \cos (x) - 9 = 0$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

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

$\frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 18 \cos (x) - 9 = 0$

Step 1.3.2.2

Separate fractions.

$\frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 18 \cos (x) - 9 = 0$

Step 1.3.2.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 18 \cos (x) - 9 = 0$

Step 1.3.2.4

Divide $2 (\sin (x))$ by $1$ .

$2 (\sin (x)) \tan (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)} - 18 \cos (x) - 9 = 0$

Step 1.3.2.5

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$2 \sin (x) \tan (x) + \tan^{2} (x) - 18 \cos (x) - 9 = 0$

Step 2

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

$x \approx 1.24904577 + 2 π n, 1.89254688 + 2 π n, \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n, 4.39063842 + 2 π n, 5.03413953 + 2 π n$ , for any integer $n$

Step 3