Trigonometry Examples

3 \sec^{2} (x) - 4 = 0

$3 \sec^{2} (x) - 4 = 0$

Step 1

Add

4

$4$ to both sides of the equation.

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

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

Step 2

Divide each term in

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

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

3

$3$ and simplify.

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Step 2.1

Divide each term in

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

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

3

$3$ .

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

3

$3$ .

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Step 2.2.1.1

Cancel the common factor.

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

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

Step 2.2.1.2

Divide

\sec^{2} (x)

$\sec^{2} (x)$ by

1

$1$ .

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

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

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

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

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

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

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

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

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}}

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

Step 4

Simplify

\pm \sqrt{\frac{4}{3}}

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

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Step 4.1

Rewrite

\sqrt{\frac{4}{3}}

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

\frac{\sqrt{4}}{\sqrt{3}}

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

\sec (x) = \pm \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

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 4.2.2

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

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

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

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

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

Step 4.3

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 4.4.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 4.4.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.4.5

Add

1

$1$ and

1

$1$ .

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

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

Step 4.4.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 4.4.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4.4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 4.4.6.4

Cancel the common factor of

2

$2$ .

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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}}}

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

Step 4.4.6.4.2

Rewrite the expression.

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

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

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

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

Step 4.4.6.5

Evaluate the exponent.

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

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

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

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

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

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

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

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

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

$\pm$ to find the first solution.

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

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

Step 5.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

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

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

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}

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

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

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

Step 6

Set up each of the solutions to solve for

x

$x$ .

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

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

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

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

Step 7

Solve for

x

$x$ in

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

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

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Step 7.1

Take the inverse secant of both sides of the equation to extract

x

$x$ from inside the secant.

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

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

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})

$arcsec (\frac{2 \sqrt{3}}{3})$ is

\frac{π}{6}

$\frac{π}{6}$ .

x = \frac{π}{6}

$x = \frac{π}{6}$

x = \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

$2 π$ to find the solution in the fourth quadrant.

x = 2 π - \frac{π}{6}

$x = 2 π - \frac{π}{6}$

Step 7.4

Simplify

2 π - \frac{π}{6}

$2 π - \frac{π}{6}$ .

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Step 7.4.1

To write

2 π

$2 π$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

x = 2 π \cdot \frac{6}{6} - \frac{π}{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

$2 π$ and

\frac{6}{6}

$\frac{6}{6}$ .

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

$x = \frac{2 π \cdot 6 - π}{6}$

x = \frac{2 π \cdot 6 - π}{6}

$x = \frac{2 π \cdot 6 - π}{6}$

Step 7.4.3

Simplify the numerator.

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Step 7.4.3.1

Multiply

6

$6$ by

2

$2$ .

x = \frac{12 π - π}{6}

$x = \frac{12 π - π}{6}$

Step 7.4.3.2

Subtract

π

$π$ from

12 π

$12 π$ .

x = \frac{11 π}{6}

$x = \frac{11 π}{6}$

x = \frac{11 π}{6}

$x = \frac{11 π}{6}$

x = \frac{11 π}{6}

$x = \frac{11 π}{6}$

Step 7.5

Find the period of

\sec (x)

$\sec (x)$ .

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Step 7.5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 7.5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 7.5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 7.5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 7.6

The period of the

\sec (x)

$\sec (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n

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

n

$n$

x = \frac{π}{6} + 2 π n, \frac{11 π}{6} + 2 π n

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

n

$n$

Step 8

Solve for

x

$x$ in

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

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

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Step 8.1

Take the inverse secant of both sides of the equation to extract

x

$x$ from inside the secant.

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

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

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})

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

\frac{5 π}{6}

$\frac{5 π}{6}$ .

x = \frac{5 π}{6}

$x = \frac{5 π}{6}$

x = \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

$2 π$ to find the solution in the third quadrant.

x = 2 π - \frac{5 π}{6}

$x = 2 π - \frac{5 π}{6}$

Step 8.4

Simplify

2 π - \frac{5 π}{6}

$2 π - \frac{5 π}{6}$ .

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Step 8.4.1

To write

2 π

$2 π$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

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

$2 π$ and

\frac{6}{6}

$\frac{6}{6}$ .

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

$x = \frac{2 π \cdot 6 - 5 π}{6}$

x = \frac{2 π \cdot 6 - 5 π}{6}

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 8.4.3

Simplify the numerator.

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Step 8.4.3.1

Multiply

6

$6$ by

2

$2$ .

x = \frac{12 π - 5 π}{6}

$x = \frac{12 π - 5 π}{6}$

Step 8.4.3.2

Subtract

5 π

$5 π$ from

12 π

$12 π$ .

x = \frac{7 π}{6}

$x = \frac{7 π}{6}$

x = \frac{7 π}{6}

$x = \frac{7 π}{6}$

x = \frac{7 π}{6}

$x = \frac{7 π}{6}$

Step 8.5

Find the period of

\sec (x)

$\sec (x)$ .

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Step 8.5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 8.5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 8.5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 8.5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 8.6

The period of the

\sec (x)

$\sec (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n

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

n

$n$

x = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n

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

n

$n$

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

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

n

$n$

Step 10

Consolidate the solutions.

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Step 10.1

Consolidate

\frac{π}{6} + 2 π n

$\frac{π}{6} + 2 π n$ and

\frac{7 π}{6} + 2 π n

$\frac{7 π}{6} + 2 π n$ to

\frac{π}{6} + π n

$\frac{π}{6} + π n$ .

x = \frac{π}{6} + π n, \frac{11 π}{6} + 2 π n, \frac{5 π}{6} + 2 π n

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

n

$n$

Step 10.2

Consolidate

\frac{11 π}{6} + 2 π n

$\frac{11 π}{6} + 2 π n$ and

\frac{5 π}{6} + 2 π n

$\frac{5 π}{6} + 2 π n$ to

\frac{5 π}{6} + π n

$\frac{5 π}{6} + π n$ .

x = \frac{π}{6} + π n, \frac{5 π}{6} + π n

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

n

$n$

x = \frac{π}{6} + π n, \frac{5 π}{6} + π n

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

n

$n$