Trigonometry Examples

$\sqrt{\frac{10}{9} + 1} = \sec (x)$

Step 1

Rewrite the equation as $\sec (x) = \sqrt{\frac{10}{9} + 1}$ .

$\sec (x) = \sqrt{\frac{10}{9} + 1}$

Step 2

Simplify $\sqrt{\frac{10}{9} + 1}$ .

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Step 2.1

Write $1$ as a fraction with a common denominator.

$\sec (x) = \sqrt{\frac{10}{9} + \frac{9}{9}}$

Step 2.2

Combine the numerators over the common denominator.

$\sec (x) = \sqrt{\frac{10 + 9}{9}}$

Step 2.3

Add $10$ and $9$ .

$\sec (x) = \sqrt{\frac{19}{9}}$

Step 2.4

Rewrite $\sqrt{\frac{19}{9}}$ as $\frac{\sqrt{19}}{\sqrt{9}}$ .

$\sec (x) = \frac{\sqrt{19}}{\sqrt{9}}$

Step 2.5

Simplify the denominator.

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Step 2.5.1

Rewrite $9$ as $3^{2}$ .

$\sec (x) = \frac{\sqrt{19}}{\sqrt{3^{2}}}$

Step 2.5.2

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

$\sec (x) = \frac{\sqrt{19}}{3}$

Step 3

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

$x = arcsec (\frac{\sqrt{19}}{3})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $arcsec (\frac{\sqrt{19}}{3})$ .

$x = 0.81172611$

Step 5

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.

$x = 2 (3.14159265) - 0.81172611$

Step 6

Simplify $2 (3.14159265) - 0.81172611$ .

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Step 6.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.81172611$

Step 6.2

Subtract $0.81172611$ from $6.2831853$ .

$x = 5.47145918$

Step 7

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

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

$x = 0.81172611 + 2 π n, 5.47145918 + 2 π n$ , for any integer $n$