Trigonometry Examples

$\frac{\csc (x)}{\cot (x)} = \sqrt{2}$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{\csc (x)}{\cot (x)}$ .

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Step 1.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\csc (x)}{\frac{\cos (x)}{\sin (x)}} = \sqrt{2}$

Step 1.1.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{\frac{1}{\sin (x)}}{\frac{\cos (x)}{\sin (x)}} = \sqrt{2}$

Step 1.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

$\frac{1}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)} = \sqrt{2}$

Step 1.1.4

Cancel the common factor of $\sin (x)$ .

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Step 1.1.4.1

Cancel the common factor.

$\frac{1}{\sin (x)} \cdot \frac{\sin (x)}{\cos (x)} = \sqrt{2}$

Step 1.1.4.2

Rewrite the expression.

$\frac{1}{\cos (x)} = \sqrt{2}$

Step 1.1.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 2

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

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

Step 3

Simplify the right side.

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Step 3.1

The exact value of $arcsec (\sqrt{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 4

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.

$x = 2 π - \frac{π}{4}$

Step 5

Simplify $2 π - \frac{π}{4}$ .

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Step 5.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.2

Combine fractions.

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Step 5.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 5.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$