Trigonometry Examples

$15 \sec^{2} (x) - 20 = 0$

Step 1

Add $20$ to both sides of the equation.

$15 \sec^{2} (x) = 20$

Step 2

Divide each term in $15 \sec^{2} (x) = 20$ by $15$ and simplify.

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Step 2.1

Divide each term in $15 \sec^{2} (x) = 20$ by $15$ .

$\frac{15 \sec^{2} (x)}{15} = \frac{20}{15}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $15$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{15 \sec^{2} (x)}{15} = \frac{20}{15}$

Step 2.2.1.2

Divide $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x) = \frac{20}{15}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $20$ and $15$ .

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Step 2.3.1.1

Factor $5$ out of $20$ .

$\sec^{2} (x) = \frac{5 (4)}{15}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $5$ out of $15$ .

$\sec^{2} (x) = \frac{5 \cdot 4}{5 \cdot 3}$

Step 2.3.1.2.2

Cancel the common factor.

$\sec^{2} (x) = \frac{5 \cdot 4}{5 \cdot 3}$

Step 2.3.1.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.4.4

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

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

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

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

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Step 4.4.6.1

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

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

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

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

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

Step 4.4.6.5

Evaluate the exponent.

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

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

Step 7

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

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Step 7.1

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

$x = arcsec (\frac{2 \sqrt{3}}{3})$

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of $arcsec (\frac{2 \sqrt{3}}{3})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 7.3

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{6}$

Step 7.4

Simplify $2 π - \frac{π}{6}$ .

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Step 7.4.1

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

$x = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.4.2

Combine fractions.

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Step 7.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - π}{6}$

Step 7.4.3

Simplify the numerator.

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Step 7.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - π}{6}$

Step 7.4.3.2

Subtract $π$ from $12 π$ .

$x = \frac{11 π}{6}$

Step 7.5

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

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Step 7.5.1

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

$\frac{2 π}{| b |}$

Step 7.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

$\frac{2 π}{1}$

Step 7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.6

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

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

Step 8

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

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Step 8.1

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

$x = arcsec (- \frac{2 \sqrt{3}}{3})$

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of $arcsec (- \frac{2 \sqrt{3}}{3})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

Step 8.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{5 π}{6}$

Step 8.4

Simplify $2 π - \frac{5 π}{6}$ .

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Step 8.4.1

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

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 8.4.2

Combine fractions.

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Step 8.4.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 8.4.3

Simplify the numerator.

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Step 8.4.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 8.4.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 8.5

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

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Step 8.5.1

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

$\frac{2 π}{| b |}$

Step 8.5.2

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

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

$\frac{2 π}{1}$

Step 8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.6

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

$x = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Step 9

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate $\frac{π}{6} + 2 π n$ and $\frac{7 π}{6} + 2 π n$ to $\frac{π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 10.2

Consolidate $\frac{11 π}{6} + 2 π n$ and $\frac{5 π}{6} + 2 π n$ to $\frac{5 π}{6} + π n$ .

$x = \frac{π}{6} + π n, \frac{5 π}{6} + π n$ , for any integer $n$