Trigonometry Examples

$9 \sec^{2} (θ) \tan (θ) = 12 \tan (θ)$

Step 1

Subtract $12 \tan (θ)$ from both sides of the equation.

$9 \sec^{2} (θ) \tan (θ) - 12 \tan (θ) = 0$

Step 2

Factor $3 \tan (θ)$ out of $9 \sec^{2} (θ) \tan (θ) - 12 \tan (θ)$ .

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Step 2.1

Factor $3 \tan (θ)$ out of $9 \sec^{2} (θ) \tan (θ)$ .

$3 \tan (θ) (3 \sec^{2} (θ)) - 12 \tan (θ) = 0$

Step 2.2

Factor $3 \tan (θ)$ out of $- 12 \tan (θ)$ .

$3 \tan (θ) (3 \sec^{2} (θ)) + 3 \tan (θ) \cdot - 4 = 0$

Step 2.3

Factor $3 \tan (θ)$ out of $3 \tan (θ) (3 \sec^{2} (θ)) + 3 \tan (θ) \cdot - 4$ .

$3 \tan (θ) (3 \sec^{2} (θ) - 4) = 0$

Step 3

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

$\tan (θ) = 0$

$3 \sec^{2} (θ) - 4 = 0$

Step 4

Set $\tan (θ)$ equal to $0$ and solve for $θ$ .

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Step 4.1

Set $\tan (θ)$ equal to $0$ .

$\tan (θ) = 0$

Step 4.2

Solve $\tan (θ) = 0$ for $θ$ .

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Step 4.2.1

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

$θ = \arctan (0)$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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

$θ = 0$

Step 4.2.3

The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the fourth quadrant.

$θ = 180 + 0$

Step 4.2.4

Add $180$ and $0$ .

$θ = 180$

Step 4.2.5

Find the period of $\tan (θ)$ .

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Step 4.2.5.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 4.2.5.2

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

$\frac{180}{| 1 |}$

Step 4.2.5.3

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

$\frac{180}{1}$

Step 4.2.5.4

Divide $180$ by $1$ .

$180$

Step 4.2.6

The period of the $\tan (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 180 n, 180 + 180 n$ , for any integer $n$

Step 5

Set $3 \sec^{2} (θ) - 4$ equal to $0$ and solve for $θ$ .

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Step 5.1

Set $3 \sec^{2} (θ) - 4$ equal to $0$ .

$3 \sec^{2} (θ) - 4 = 0$

Step 5.2

Solve $3 \sec^{2} (θ) - 4 = 0$ for $θ$ .

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Step 5.2.1

Add $4$ to both sides of the equation.

$3 \sec^{2} (θ) = 4$

Step 5.2.2

Divide each term in $3 \sec^{2} (θ) = 4$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 \sec^{2} (θ) = 4$ by $3$ .

$\frac{3 \sec^{2} (θ)}{3} = \frac{4}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 \sec^{2} (θ)}{3} = \frac{4}{3}$

Step 5.2.2.2.1.2

Divide $\sec^{2} (θ)$ by $1$ .

$\sec^{2} (θ) = \frac{4}{3}$

Step 5.2.3

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

$\sec (θ) = \pm \sqrt{\frac{4}{3}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{4}{3}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$\sec (θ) = \pm \frac{\sqrt{4}}{\sqrt{3}}$

Step 5.2.4.2

Simplify the numerator.

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Step 5.2.4.2.1

Rewrite $4$ as $2^{2}$ .

$\sec (θ) = \pm \frac{\sqrt{2^{2}}}{\sqrt{3}}$

Step 5.2.4.2.2

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

$\sec (θ) = \pm \frac{2}{\sqrt{3}}$

Step 5.2.4.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sec (θ) = \pm \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.2.4.4

Combine and simplify the denominator.

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Step 5.2.4.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.2.4.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.2.4.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.2.4.4.4

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

$\sec (θ) = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.2.4.4.5

Add $1$ and $1$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.2.4.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.2.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sec (θ) = \pm \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.6.4.1

Cancel the common factor.

$\sec (θ) = \pm \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.2.4.4.6.4.2

Rewrite the expression.

$\sec (θ) = \pm \frac{2 \sqrt{3}}{3^{1}}$

Step 5.2.4.4.6.5

Evaluate the exponent.

$\sec (θ) = \pm \frac{2 \sqrt{3}}{3}$

Step 5.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

Step 5.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (θ) = - \frac{2 \sqrt{3}}{3}$

Step 5.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (θ) = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}$

Step 5.2.6

Set up each of the solutions to solve for $θ$ .

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

$\sec (θ) = - \frac{2 \sqrt{3}}{3}$

Step 5.2.7

Solve for $θ$ in $\sec (θ) = \frac{2 \sqrt{3}}{3}$ .

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Step 5.2.7.1

Take the inverse secant of both sides of the equation to extract $θ$ from inside the secant.

$θ = arcsec (\frac{2 \sqrt{3}}{3})$

Step 5.2.7.2

Simplify the right side.

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Step 5.2.7.2.1

The exact value of $arcsec (\frac{2 \sqrt{3}}{3})$ is $30$ .

$θ = 30$

Step 5.2.7.3

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$θ = 360 - 30$

Step 5.2.7.4

Subtract $30$ from $360$ .

$θ = 330$

Step 5.2.7.5

Find the period of $\sec (θ)$ .

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Step 5.2.7.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 5.2.7.5.2

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

$\frac{360}{| 1 |}$

Step 5.2.7.5.3

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

$\frac{360}{1}$

Step 5.2.7.5.4

Divide $360$ by $1$ .

$360$

Step 5.2.7.6

The period of the $\sec (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 30 + 360 n, 330 + 360 n$ , for any integer $n$

Step 5.2.8

Solve for $θ$ in $\sec (θ) = - \frac{2 \sqrt{3}}{3}$ .

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Step 5.2.8.1

Take the inverse secant of both sides of the equation to extract $θ$ from inside the secant.

$θ = arcsec (- \frac{2 \sqrt{3}}{3})$

Step 5.2.8.2

Simplify the right side.

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Step 5.2.8.2.1

The exact value of $arcsec (- \frac{2 \sqrt{3}}{3})$ is $150$ .

$θ = 150$

Step 5.2.8.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the third quadrant.

$θ = 360 - 150$

Step 5.2.8.4

Subtract $150$ from $360$ .

$θ = 210$

Step 5.2.8.5

Find the period of $\sec (θ)$ .

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Step 5.2.8.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 5.2.8.5.2

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

$\frac{360}{| 1 |}$

Step 5.2.8.5.3

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

$\frac{360}{1}$

Step 5.2.8.5.4

Divide $360$ by $1$ .

$360$

Step 5.2.8.6

The period of the $\sec (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 150 + 360 n, 210 + 360 n$ , for any integer $n$

Step 5.2.9

List all of the solutions.

$θ = 30 + 360 n, 330 + 360 n, 150 + 360 n, 210 + 360 n$ , for any integer $n$

Step 5.2.10

Consolidate the solutions.

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Step 5.2.10.1

Consolidate $30 + 360 n$ and $210 + 360 n$ to $30 + 180 n$ .

$θ = 30 + 180 n, 330 + 360 n, 150 + 360 n$ , for any integer $n$

Step 5.2.10.2

Consolidate $330 + 360 n$ and $150 + 360 n$ to $150 + 180 n$ .

$θ = 30 + 180 n, 150 + 180 n$ , for any integer $n$

Step 6

The final solution is all the values that make $3 \tan (θ) (3 \sec^{2} (θ) - 4) = 0$ true.

$θ = 180 n, 180 + 180 n, 30 + 180 n, 150 + 180 n$ , for any integer $n$

Step 7

Consolidate $180 n$ and $180 + 180 n$ to $180 n$ .

$θ = 180 n, 30 + 180 n, 150 + 180 n$ , for any integer $n$