Trigonometry Examples

$\sqrt{\frac{\cos (x)}{\tan (x)}} = \cot (x)$

Step 1

Subtract $\cot (x)$ from both sides of the equation.

$\sqrt{\frac{\cos (x)}{\tan (x)}} - \cot (x) = 0$

Step 2

Simplify $\sqrt{\frac{\cos (x)}{\tan (x)}} - \cot (x)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\sqrt{\frac{\cos (x)}{\frac{\sin (x)}{\cos (x)}}} - \cot (x) = 0$

Step 2.1.2

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

$\sqrt{\cos (x) \frac{\cos (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.3

Write $\cos (x)$ as a fraction with denominator $1$ .

$\sqrt{\frac{\cos (x)}{1} \cdot \frac{\cos (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

$\sqrt{\cos (x) \frac{\cos (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.4.2

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

$\sqrt{\frac{\cos (x) \cos (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

Raise $\cos (x)$ to the power of $1$ .

$\sqrt{\frac{\cos^{1} (x) \cos (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.5.2

Raise $\cos (x)$ to the power of $1$ .

$\sqrt{\frac{\cos^{1} (x) \cos^{1} (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{\frac{{\cos (x)}^{1 + 1}}{\sin (x)}} - \cot (x) = 0$

Step 2.1.5.4

Add $1$ and $1$ .

$\sqrt{\frac{\cos^{2} (x)}{\sin (x)}} - \cot (x) = 0$

Step 2.1.6

Rewrite $\sqrt{\frac{\cos^{2} (x)}{\sin (x)}}$ as $\frac{\sqrt{\cos^{2} (x)}}{\sqrt{\sin (x)}}$ .

$\frac{\sqrt{\cos^{2} (x)}}{\sqrt{\sin (x)}} - \cot (x) = 0$

Step 2.1.7

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

$\frac{\cos (x)}{\sqrt{\sin (x)}} - \cot (x) = 0$

Step 2.1.8

Multiply $\frac{\cos (x)}{\sqrt{\sin (x)}}$ by $\frac{\sqrt{\sin (x)}}{\sqrt{\sin (x)}}$ .

$\frac{\cos (x)}{\sqrt{\sin (x)}} \cdot \frac{\sqrt{\sin (x)}}{\sqrt{\sin (x)}} - \cot (x) = 0$

Step 2.1.9

Combine and simplify the denominator.

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Step 2.1.9.1

Multiply $\frac{\cos (x)}{\sqrt{\sin (x)}}$ by $\frac{\sqrt{\sin (x)}}{\sqrt{\sin (x)}}$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{\sqrt{\sin (x)} \sqrt{\sin (x)}} - \cot (x) = 0$

Step 2.1.9.2

Raise $\sqrt{\sin (x)}$ to the power of $1$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sqrt{\sin (x)}}^{1} \sqrt{\sin (x)}} - \cot (x) = 0$

Step 2.1.9.3

Raise $\sqrt{\sin (x)}$ to the power of $1$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sqrt{\sin (x)}}^{1} {\sqrt{\sin (x)}}^{1}} - \cot (x) = 0$

Step 2.1.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sqrt{\sin (x)}}^{1 + 1}} - \cot (x) = 0$

Step 2.1.9.5

Add $1$ and $1$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sqrt{\sin (x)}}^{2}} - \cot (x) = 0$

Step 2.1.9.6

Rewrite ${\sqrt{\sin (x)}}^{2}$ as $\sin (x)$ .

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Step 2.1.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sin (x)}$ as ${\sin (x)}^{\frac{1}{2}}$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{{({\sin (x)}^{\frac{1}{2}})}^{2}} - \cot (x) = 0$

Step 2.1.9.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sin (x)}^{\frac{1}{2} \cdot 2}} - \cot (x) = 0$

Step 2.1.9.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sin (x)}^{\frac{2}{2}}} - \cot (x) = 0$

Step 2.1.9.6.4

Cancel the common factor of $2$ .

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Step 2.1.9.6.4.1

Cancel the common factor.

$\frac{\cos (x) \sqrt{\sin (x)}}{{\sin (x)}^{\frac{2}{2}}} - \cot (x) = 0$

Step 2.1.9.6.4.2

Rewrite the expression.

$\frac{\cos (x) \sqrt{\sin (x)}}{\sin^{1} (x)} - \cot (x) = 0$

Step 2.1.9.6.5

Simplify.

$\frac{\cos (x) \sqrt{\sin (x)}}{\sin (x)} - \cot (x) = 0$

Step 2.1.10

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

$\frac{\cos (x) \sqrt{\sin (x)}}{\sin (x)} - \frac{\cos (x)}{\sin (x)} = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Separate fractions.

$\frac{\sqrt{\sin (x)}}{1} \cdot \frac{\cos (x)}{\sin (x)} - \frac{\cos (x)}{\sin (x)} = 0$

Step 2.2.2

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

$\frac{\sqrt{\sin (x)}}{1} \cot (x) - \frac{\cos (x)}{\sin (x)} = 0$

Step 2.2.3

Divide $\sqrt{\sin (x)}$ by $1$ .

$\sqrt{\sin (x)} \cot (x) - \frac{\cos (x)}{\sin (x)} = 0$

Step 2.2.4

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

$\sqrt{\sin (x)} \cot (x) - \cot (x) = 0$

Step 3

Factor $\sqrt{\sin (x)} \cot (x) - \cot (x)$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sin (x)}$ as ${\sin (x)}^{\frac{1}{2}}$ .

${\sin (x)}^{\frac{1}{2}} \cot (x) - \cot (x) = 0$

Step 3.2

Factor $\cot (x)$ out of ${\sin (x)}^{\frac{1}{2}} \cot (x) - \cot (x)$ .

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Step 3.2.1

Factor $\cot (x)$ out of ${\sin (x)}^{\frac{1}{2}} \cot (x)$ .

$\cot (x) {\sin (x)}^{\frac{1}{2}} - \cot (x) = 0$

Step 3.2.2

Factor $\cot (x)$ out of $- \cot (x)$ .

$\cot (x) {\sin (x)}^{\frac{1}{2}} + \cot (x) \cdot - 1 = 0$

Step 3.2.3

Factor $\cot (x)$ out of $\cot (x) {\sin (x)}^{\frac{1}{2}} + \cot (x) \cdot - 1$ .

$\cot (x) ({\sin (x)}^{\frac{1}{2}} - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cot (x) = 0$

${\sin (x)}^{\frac{1}{2}} - 1 = 0$

Step 5

Set $\cot (x)$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\cot (x)$ equal to $0$ .

$\cot (x) = 0$

Step 5.2

Solve $\cot (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = arccot (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of $arccot (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 5.2.3

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.

$x = π + \frac{π}{2}$

Step 5.2.4

Simplify $π + \frac{π}{2}$ .

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Step 5.2.4.1

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

$x = π \cdot \frac{2}{2} + \frac{π}{2}$

Step 5.2.4.2

Combine fractions.

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Step 5.2.4.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} + \frac{π}{2}$

Step 5.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 5.2.4.3

Simplify the numerator.

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Step 5.2.4.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 5.2.4.3.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 5.2.5

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

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Step 5.2.5.1

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

$\frac{π}{| b |}$

Step 5.2.5.2

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

$\frac{π}{| 1 |}$

Step 5.2.5.3

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

$\frac{π}{1}$

Step 5.2.5.4

Divide $π$ by $1$ .

$π$

Step 5.2.6

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

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n$ , for any integer $n$

Step 6

Set ${\sin (x)}^{\frac{1}{2}} - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set ${\sin (x)}^{\frac{1}{2}} - 1$ equal to $0$ .

${\sin (x)}^{\frac{1}{2}} - 1 = 0$

Step 6.2

Solve ${\sin (x)}^{\frac{1}{2}} - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

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

Step 6.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 6.2.3

Simplify the exponent.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify ${({\sin (x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.3.1.1.1

Multiply the exponents in ${({\sin (x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 6.2.3.1.1.1.2.2

Rewrite the expression.

$\sin^{1} (x) = 1^{2}$

Step 6.2.3.1.1.2

Simplify.

$\sin (x) = 1^{2}$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

One to any power is one.

$\sin (x) = 1$

Step 6.2.4

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

$x = \arcsin (1)$

Step 6.2.5

Simplify the right side.

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Step 6.2.5.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 6.2.6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - \frac{π}{2}$

Step 6.2.7

Simplify $π - \frac{π}{2}$ .

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Step 6.2.7.1

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

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

Step 6.2.7.2

Combine fractions.

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Step 6.2.7.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.2.7.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 6.2.7.3

Simplify the numerator.

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Step 6.2.7.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 6.2.7.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 6.2.8

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

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Step 6.2.8.1

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

$\frac{2 π}{| b |}$

Step 6.2.8.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.8.3

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

$\frac{2 π}{1}$

Step 6.2.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.9

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

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

Step 7

The final solution is all the values that make $\cot (x) ({\sin (x)}^{\frac{1}{2}} - 1) = 0$ true.

$x = \frac{π}{2} + π n, \frac{3 π}{2} + π n, \frac{π}{2} + 2 π n$ , for any integer $n$

Step 8

Consolidate the answers.

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