Chemistry Examples

$\sin (- x) \cot (x) = \cos (x)$

Step 1

Simplify the left side.

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Step 1.1

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

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Step 1.1.1

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$- \sin (x) \cot (x) = \cos (x)$

Step 1.1.2

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

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Step 1.1.2.1

Add parentheses.

$- (\sin (x) \cot (x)) = \cos (x)$

Step 1.1.2.2

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

$- (\cot (x) \sin (x)) = \cos (x)$

Step 1.1.2.3

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

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

Step 1.1.2.4

Cancel the common factors.

$- \cos (x) = \cos (x)$

Step 2

Move all terms containing $\cos (x)$ to the left side of the equation.

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Step 2.1

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

$- \cos (x) - \cos (x) = 0$

Step 2.2

Subtract $\cos (x)$ from $- \cos (x)$ .

$- 2 \cos (x) = 0$

Step 3

Divide each term in $- 2 \cos (x) = 0$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 \cos (x) = 0$ by $- 2$ .

$\frac{- 2 \cos (x)}{- 2} = \frac{0}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 \cos (x)}{- 2} = \frac{0}{- 2}$

Step 3.2.1.2

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

$\cos (x) = \frac{0}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 2$ .

$\cos (x) = 0$

Step 4

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

$x = \arccos (0)$

Step 5

Simplify the right side.

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Step 5.1

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

$x = \frac{π}{2}$

Step 6

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.

$x = 2 π - \frac{π}{2}$

Step 7

Simplify $2 π - \frac{π}{2}$ .

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Step 7.1

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

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

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 7.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 8

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

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 1 |}$

Step 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 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

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

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

Step 10

Consolidate the answers.

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