Trigonometry Examples

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

Step 1

Simplify the left side of the equation.

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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.

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

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

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

Step 1.1.4.3

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

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

Step 1.1.4.4

Add $1$ and $2$ .

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Factor $\sin^{2} (x)$ out of $\sin^{3} (x)$ .

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

Step 1.2.2

Multiply by $1$ .

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

Step 1.2.3

Separate fractions.

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply by $1$ .

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

Step 2.3

Factor $\tan^{2} (x)$ out of $\tan^{2} (x) (2 \sin (x)) + \tan^{2} (x) \cdot 1$ .

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

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan^{2} (x) = 0$

$2 \sin (x) + 1 = 0$

Step 4

Set $\tan^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\tan^{2} (x)$ equal to $0$ .

$\tan^{2} (x) = 0$

Step 4.2

Solve $\tan^{2} (x) = 0$ for $x$ .

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Step 4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\tan (x) = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$\tan (x) = \pm \sqrt{0^{2}}$

Step 4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\tan (x) = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$\tan (x) = 0$

Step 4.2.3

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

$x = \arctan (0)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The exact value of $\arctan (0)$ is $0$ .

$x = 0$

Step 4.2.5

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + 0$

Step 4.2.6

Add $π$ and $0$ .

$x = π$

Step 4.2.7

Find the period of $\tan (x)$ .

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Step 4.2.7.1

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

$\frac{π}{| b |}$

Step 4.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 4.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 4.2.7.4

Divide $π$ by $1$ .

$π$

Step 4.2.8

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

$x = π n, π + π n$ , for any integer $n$

Step 5

Set $2 \sin (x) + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 \sin (x) + 1$ equal to $0$ .

$2 \sin (x) + 1 = 0$

Step 5.2

Solve $2 \sin (x) + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$2 \sin (x) = - 1$

Step 5.2.2

Divide each term in $2 \sin (x) = - 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 \sin (x) = - 1$ by $2$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

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

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

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

Step 5.2.3

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

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

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

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

$x = - \frac{π}{6}$

Step 5.2.5

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.

$x = 2 π + \frac{π}{6} + π$

Step 5.2.6

Simplify the expression to find the second solution.

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Step 5.2.6.1

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

$x = 2 π + \frac{π}{6} + π - 2 π$

Step 5.2.6.2

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

$x = \frac{7 π}{6}$

Step 5.2.7

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

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Step 5.2.7.1

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

$\frac{2 π}{| b |}$

Step 5.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8

Add $2 π$ to every negative angle to get positive angles.

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Step 5.2.8.1

Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

Step 5.2.8.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 5.2.8.3

Combine fractions.

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Step 5.2.8.3.1

Combine $2 π$ and $\frac{6}{6}$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 5.2.8.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Step 5.2.8.4

Simplify the numerator.

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Step 5.2.8.4.1

Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Step 5.2.8.4.2

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

Step 5.2.8.5

List the new angles.

$x = \frac{11 π}{6}$

Step 5.2.9

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $\tan^{2} (x) (2 \sin (x) + 1) = 0$ true.

$x = π n, π + π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

Step 7

Consolidate $π n$ and $π + π n$ to $π n$ .

$x = π n, \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$