Solve for θ in Degrees 16sec(theta)^2-25=0

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Trigonometry

$16 \sec^{2} (θ) - 25 = 0$

Add $25$ to both sides of the equation.

$16 \sec^{2} (θ) = 25$

Divide each term in $16 \sec^{2} (θ) = 25$ by $16$ and simplify.

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Divide each term in $16 \sec^{2} (θ) = 25$ by $16$ .

$\frac{16 \sec^{2} (θ)}{16} = \frac{25}{16}$

Simplify the left side.

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Cancel the common factor of $16$ .

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Cancel the common factor.

$\frac{16 \sec^{2} (θ)}{16} = \frac{25}{16}$

Divide $\sec^{2} (θ)$ by $1$ .

$\sec^{2} (θ) = \frac{25}{16}$

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

$\sec (θ) = \pm \sqrt{\frac{25}{16}}$

Simplify $\pm \sqrt{\frac{25}{16}}$ .

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Rewrite $\sqrt{\frac{25}{16}}$ as $\frac{\sqrt{25}}{\sqrt{16}}$ .

$\sec (θ) = \pm \frac{\sqrt{25}}{\sqrt{16}}$

Simplify the numerator.

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Rewrite $25$ as $5^{2}$ .

$\sec (θ) = \pm \frac{\sqrt{5^{2}}}{\sqrt{16}}$

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

$\sec (θ) = \pm \frac{5}{\sqrt{16}}$

Simplify the denominator.

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Rewrite $16$ as $4^{2}$ .

$\sec (θ) = \pm \frac{5}{\sqrt{4^{2}}}$

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

$\sec (θ) = \pm \frac{5}{4}$

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

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

$\sec (θ) = \frac{5}{4}$

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

$\sec (θ) = - \frac{5}{4}$

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

$\sec (θ) = \frac{5}{4}, - \frac{5}{4}$

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

$\sec (θ) = \frac{5}{4}$

$\sec (θ) = - \frac{5}{4}$

Solve for $θ$ in $\sec (θ) = \frac{5}{4}$ .

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

$θ = arcsec (\frac{5}{4})$

Simplify the right side.

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Evaluate $arcsec (\frac{5}{4})$ .

$θ = 36.86989764$

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.

$θ = 360 - 36.86989764$

Subtract $36.86989764$ from $360$ .

$θ = 323.13010235$

Find the period of $\sec (θ)$ .

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

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

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

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

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

$\frac{360}{1}$

Divide $360$ by $1$ .

$360$

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

$θ = 36.86989764 + 360 n, 323.13010235 + 360 n$ , for any integer $n$

Solve for $θ$ in $\sec (θ) = - \frac{5}{4}$ .

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

$θ = arcsec (- \frac{5}{4})$

Simplify the right side.

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Evaluate $arcsec (- \frac{5}{4})$ .

$θ = 143.13010235$

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.

$θ = 360 - 143.13010235$

Subtract $143.13010235$ from $360$ .

$θ = 216.86989764$

Find the period of $\sec (θ)$ .

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

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

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

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

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

$\frac{360}{1}$

Divide $360$ by $1$ .

$360$

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

$θ = 143.13010235 + 360 n, 216.86989764 + 360 n$ , for any integer $n$

List all of the solutions.

$θ = 36.86989764 + 360 n, 323.13010235 + 360 n, 143.13010235 + 360 n, 216.86989764 + 360 n$ , for any integer $n$

Consolidate the solutions.

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Consolidate $36.86989764 + 360 n$ and $216.86989764 + 360 n$ to $36.86989764 + 180 n$ .

$θ = 36.86989764 + 180 n, 323.13010235 + 360 n, 143.13010235 + 360 n$ , for any integer $n$

Consolidate $323.13010235 + 360 n$ and $143.13010235 + 360 n$ to $143.13010235 + 180 n$ .

$θ = 36.86989764 + 180 n, 143.13010235 + 180 n$ , for any integer $n$

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