Trigonometry Examples

$6 \tan^{2} (x) \cos (x) + 3 \tan^{2} (x) = 0$

Step 1

Simplify $6 \tan^{2} (x) \cos (x) + 3 \tan^{2} (x)$ .

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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.

$6 {(\frac{\sin (x)}{\cos (x)})}^{2} \cos (x) + 3 \tan^{2} (x) = 0$

Step 1.1.2

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

$6 \frac{\sin^{2} (x)}{\cos^{2} (x)} \cos (x) + 3 \tan^{2} (x) = 0$

Step 1.1.3

Combine $6$ and $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$\frac{6 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + 3 \tan^{2} (x) = 0$

Step 1.1.4

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

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Step 1.1.4.1

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

$\frac{6 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + 3 \tan^{2} (x) = 0$

Step 1.1.4.2

Cancel the common factor.

$\frac{6 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + 3 \tan^{2} (x) = 0$

Step 1.1.4.3

Rewrite the expression.

$\frac{6 \sin^{2} (x)}{\cos (x)} + 3 \tan^{2} (x) = 0$

Step 1.1.5

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

$\frac{6 \sin^{2} (x)}{\cos (x)} + 3 {(\frac{\sin (x)}{\cos (x)})}^{2} = 0$

Step 1.1.6

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

$\frac{6 \sin^{2} (x)}{\cos (x)} + 3 \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.1.7

Combine $3$ and $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$\frac{6 \sin^{2} (x)}{\cos (x)} + \frac{3 \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

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

$\frac{6 (\sin (x) \sin (x))}{\cos (x)} + \frac{3 \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.2.2

Separate fractions.

$\frac{6 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{3 \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.2.3

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

$\frac{6 (\sin (x))}{1} \tan (x) + \frac{3 \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.2.4

Divide $6 (\sin (x))$ by $1$ .

$6 (\sin (x)) \tan (x) + \frac{3 \sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.2.5

Multiply by $1$ .

$6 \sin (x) \tan (x) + \frac{3 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)} = 0$

Step 1.2.6

Multiply by $1$ .

$6 \sin (x) \tan (x) + \frac{3 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1} = 0$

Step 1.2.7

Separate fractions.

$6 \sin (x) \tan (x) + \frac{3 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} = 0$

Step 1.2.8

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

$6 \sin (x) \tan (x) + \frac{3 \cdot (1)}{1} \tan^{2} (x) = 0$

Step 1.2.9

Multiply $3$ by $1$ .

$6 \sin (x) \tan (x) + \frac{3}{1} \tan^{2} (x) = 0$

Step 1.2.10

Divide $3$ by $1$ .

$6 \sin (x) \tan (x) + 3 \tan^{2} (x) = 0$

Step 2

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

$x = 2 π n, \frac{2 π}{3} + 2 π n, π + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 3