Trigonometry Examples

$\sqrt{3} \sec (θ) + 2 = 0$ , $[0, 2 π)$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{3} \sec (θ) = - 2$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $\sec (θ)$ by $1$ .

$\sec (θ) = \frac{- 2}{\sqrt{3}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\sec (θ) = - \frac{2}{\sqrt{3}}$

Step 2.3.2

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

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

Step 2.3.3

Combine and simplify the denominator.

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

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

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

Step 2.3.3.4

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

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

Step 2.3.3.5

Add $1$ and $1$ .

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

Step 2.3.3.6

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

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Step 2.3.3.6.1

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

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

Step 2.3.3.6.2

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

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

Step 2.3.3.6.3

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

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

Step 2.3.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.3.6.4.1

Cancel the common factor.

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

Step 2.3.3.6.4.2

Rewrite the expression.

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

Step 2.3.3.6.5

Evaluate the exponent.

$\sec (θ) = - \frac{2 \sqrt{3}}{3}$

Step 3

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 4

Simplify the right side.

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Step 4.1

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

$θ = \frac{5 π}{6}$

Step 5

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.

$θ = 2 π - \frac{5 π}{6}$

Step 6

Simplify $2 π - \frac{5 π}{6}$ .

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Step 6.1

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

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

Step 6.2

Combine fractions.

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Step 6.2.1

Combine $2 π$ and $\frac{6}{6}$ .

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

Step 6.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 6 - 5 π}{6}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Multiply $6$ by $2$ .

$θ = \frac{12 π - 5 π}{6}$

Step 6.3.2

Subtract $5 π$ from $12 π$ .

$θ = \frac{7 π}{6}$

Step 7

Find the period of $\sec (θ)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 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 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

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

Step 9

Find the values of $n$ that produce a value within the interval $[0, 2 π)$ .

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Step 9.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 9.1.1

Plug in $0$ for $n$ .

$\frac{5 π}{6} + 2 π (0)$

Step 9.1.2

Simplify.

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Step 9.1.2.1

Multiply $2 π (0)$ .

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Step 9.1.2.1.1

Multiply $0$ by $2$ .

$\frac{5 π}{6} + 0 π$

Step 9.1.2.1.2

Multiply $0$ by $π$ .

$\frac{5 π}{6} + 0$

Step 9.1.2.2

Add $\frac{5 π}{6}$ and $0$ .

$\frac{5 π}{6}$

Step 9.1.3

The interval $[0, 2 π)$ contains $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Step 9.2

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 9.2.1

Plug in $0$ for $n$ .

$\frac{7 π}{6} + 2 π (0)$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Multiply $2 π (0)$ .

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Step 9.2.2.1.1

Multiply $0$ by $2$ .

$\frac{7 π}{6} + 0 π$

Step 9.2.2.1.2

Multiply $0$ by $π$ .

$\frac{7 π}{6} + 0$

Step 9.2.2.2

Add $\frac{7 π}{6}$ and $0$ .

$\frac{7 π}{6}$

Step 9.2.3

The interval $[0, 2 π)$ contains $\frac{7 π}{6}$ .

$θ = \frac{5 π}{6}, \frac{7 π}{6}$