Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify $\sin (x) \cot (x) \csc (x)$ .

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Step 1.1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1.1

Reorder $\sin (x)$ and $\cot (x)$ .

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Cancel the common factors.

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

Step 1.1.2

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

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

Step 1.1.3

Combine $\cos (x)$ and $\frac{1}{\sin (x)}$ .

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

Step 1.1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 2

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

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

Step 3

Simplify the right side.

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Step 3.1

Evaluate $arccot (\sqrt{2})$ .

$x = 0.6154797$

Step 4

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = (3.14159265) + 0.6154797$

Step 5

Solve for $x$ .

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Step 5.1

Remove parentheses.

$x = 3.14159265 + 0.6154797$

Step 5.2

Remove parentheses.

$x = (3.14159265) + 0.6154797$

Step 5.3

Add $3.14159265$ and $0.6154797$ .

$x = 3.75707236$

Step 6

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

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Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

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

$\frac{π}{| 1 |}$

Step 6.3

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

$\frac{π}{1}$

Step 6.4

Divide $π$ by $1$ .

$π$

Step 7

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

$x = 0.6154797 + π n, 3.75707236 + π n$ , for any integer $n$

Step 8

Consolidate $0.6154797 + π n$ and $3.75707236 + π n$ to $0.6154797 + π n$ .

$x = 0.6154797 + π n$ , for any integer $n$