Solve for x 1/(cos(x))=8/3

$\frac{1}{\cos (x)} = \frac{8}{3}$

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 \cdot 3 = \cos (x) \cdot 8$

Solve the equation for $\cos (x)$ .

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Rewrite the equation as $\cos (x) \cdot 8 = 1 \cdot 3$ .

$\cos (x) \cdot 8 = 1 \cdot 3$

Multiply $3$ by $1$ .

$\cos (x) \cdot 8 = 3$

Divide each term by $8$ and simplify.

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Divide each term in $\cos (x) \cdot 8 = 3$ by $8$ .

$\frac{\cos (x) \cdot 8}{8} = \frac{3}{8}$

Cancel the common factor of $8$ .

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Cancel the common factor.

$\frac{\cos (x) \cdot 8}{8} = \frac{3}{8}$

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{3}{8}$

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

$x = \arccos (\frac{3}{8})$

Evaluate $\arccos (\frac{3}{8})$ .

$x = 1.18639955$

The cosine 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.

$x = 2 (3.14159265) - 1.18639955$

Simplify $2 (3.14159265) - 1.18639955$ .

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Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.18639955$

Subtract $1.18639955$ from $6.2831853$ .

$x = 5.09678575$

Find the period.

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

Solve the equation.

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$x = 1.18639955 + 2 π n, 5.09678575 + 2 π n$ , for any integer $n$

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