Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

$- \frac{\cos^{1} (x) \cos (x)}{\sin (x)} + \sin (- x)$

Step 1.4.3

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

$- \frac{\cos^{1} (x) \cos^{1} (x)}{\sin (x)} + \sin (- x)$

Step 1.4.4

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

$- \frac{{\cos (x)}^{1 + 1}}{\sin (x)} + \sin (- x)$

Step 1.4.5

Add $1$ and $1$ .

$- \frac{\cos^{2} (x)}{\sin (x)} + \sin (- x)$

Step 1.5

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

$- \frac{\cos^{2} (x)}{\sin (x)} - \sin (x)$

Step 2

Simplify each term.

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Step 2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 2.2

Separate fractions.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 4

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 5

Substitute the actual values of $a = - \cos (x) \cot (x)$ and $b = - 1 \sin (x)$ .

$| z | = \sqrt{{(- 1 \sin (x))}^{2} + {(- \cos (x) \cot (x))}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$| z | = \sqrt{{(- \sin (x))}^{2} + {(- \cos (x) \cot (x))}^{2}}$

Step 6.2

Apply the product rule to $- \sin (x)$ .

$| z | = \sqrt{{(- 1)}^{2} \sin^{2} (x) + {(- \cos (x) \cot (x))}^{2}}$

Step 6.3

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1 \sin^{2} (x) + {(- \cos (x) \cot (x))}^{2}}$

Step 6.4

Multiply $\sin^{2} (x)$ by $1$ .

$| z | = \sqrt{\sin^{2} (x) + {(- \cos (x) \cot (x))}^{2}}$

Step 6.5

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

$| z | = \sqrt{\sin^{2} (x) + {(- \cos (x) \frac{\cos (x)}{\sin (x)})}^{2}}$

Step 6.6

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

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Step 6.6.1

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

$| z | = \sqrt{\sin^{2} (x) + {(- \frac{\cos (x) \cos (x)}{\sin (x)})}^{2}}$

Step 6.6.2

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

$| z | = \sqrt{\sin^{2} (x) + {(- \frac{\cos (x) \cos (x)}{\sin (x)})}^{2}}$

Step 6.6.3

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

$| z | = \sqrt{\sin^{2} (x) + {(- \frac{\cos (x) \cos (x)}{\sin (x)})}^{2}}$

Step 6.6.4

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

$| z | = \sqrt{\sin^{2} (x) + {(- \frac{{\cos (x)}^{1 + 1}}{\sin (x)})}^{2}}$

Step 6.6.5

Add $1$ and $1$ .

$| z | = \sqrt{\sin^{2} (x) + {(- \frac{\cos^{2} (x)}{\sin (x)})}^{2}}$

Step 6.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.7.1

Apply the product rule to $- \frac{\cos^{2} (x)}{\sin (x)}$ .

$| z | = \sqrt{\sin^{2} (x) + {(- 1)}^{2} {(\frac{\cos^{2} (x)}{\sin (x)})}^{2}}$

Step 6.7.2

Apply the product rule to $\frac{\cos^{2} (x)}{\sin (x)}$ .

$| z | = \sqrt{\sin^{2} (x) + {(- 1)}^{2} (\frac{{(\cos^{2} (x))}^{2}}{\sin^{2} (x)})}$

Step 6.8

Simplify the expression.

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Step 6.8.1

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{\sin^{2} (x) + 1 (\frac{{(\cos^{2} (x))}^{2}}{\sin^{2} (x)})}$

Step 6.8.2

Multiply $\frac{{(\cos^{2} (x))}^{2}}{\sin^{2} (x)}$ by $1$ .

$| z | = \sqrt{\sin^{2} (x) + \frac{{(\cos^{2} (x))}^{2}}{\sin^{2} (x)}}$

Step 6.8.3

Multiply the exponents in ${(\cos^{2} (x))}^{2}$ .

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Step 6.8.3.1

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

$| z | = \sqrt{\sin^{2} (x) + \frac{{\cos (x)}^{2 \cdot 2}}{\sin^{2} (x)}}$

Step 6.8.3.2

Multiply $2$ by $2$ .

$| z | = \sqrt{\sin^{2} (x) + \frac{\cos^{4} (x)}{\sin^{2} (x)}}$

Step 6.9

Simplify each term.

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Step 6.9.1

Factor $\cos^{2} (x)$ out of $\cos^{4} (x)$ .

$| z | = \sqrt{\sin^{2} (x) + \frac{\cos^{2} (x) \cos^{2} (x)}{\sin^{2} (x)}}$

Step 6.9.2

Multiply by $1$ .

$| z | = \sqrt{\sin^{2} (x) + \frac{\cos^{2} (x) \cos^{2} (x)}{\sin^{2} (x) \cdot 1}}$

Step 6.9.3

Separate fractions.

$| z | = \sqrt{\sin^{2} (x) + \frac{\cos^{2} (x)}{1} \cdot \frac{\cos^{2} (x)}{\sin^{2} (x)}}$

Step 6.9.4

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

$| z | = \sqrt{\sin^{2} (x) + \frac{\cos^{2} (x)}{1} \cdot \cot^{2} (x)}$

Step 6.9.5

Divide $\cos^{2} (x)$ by $1$ .

$| z | = \sqrt{\sin^{2} (x) + \cos^{2} (x) \cot^{2} (x)}$

Step 7

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- 1 \sin (x)}{- \cos (x) \cot (x)})$

Step 8

Substitute the values of $θ = \arctan (\frac{- 1 \sin (x)}{- \cos (x) \cot (x)})$ and $| z | = \sqrt{\sin^{2} (x) + \cos^{2} (x) \cot^{2} (x)}$ .

$\sqrt{\sin^{2} (x) + \cos^{2} (x) \cot^{2} (x)} (\cos (\arctan (\frac{- 1 \sin (x)}{- \cos (x) \cot (x)})) + i \sin (\arctan (\frac{- 1 \sin (x)}{- \cos (x) \cot (x)})))$