Calculus Examples

$y = 3 + \cot (x) - 2 \csc (x)$

Step 1

Set $y$ as a function of $x$ .

$f (x) = 3 + \cot (x) - 2 \csc (x)$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $3 + \cot (x) - 2 \csc (x)$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- 2 \csc (x)]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- 2 \csc (x)]$

Step 2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- 2 \csc (x)]$

Step 2.2

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$0 - \csc^{2} (x) + \frac{d}{d x} [- 2 \csc (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [- 2 \csc (x)]$ .

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Step 2.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \csc (x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\csc (x)]$ .

$0 - \csc^{2} (x) - 2 \frac{d}{d x} [\csc (x)]$

Step 2.3.2

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$0 - \csc^{2} (x) - 2 (- \csc (x) \cot (x))$

Step 2.3.3

Multiply $- 1$ by $- 2$ .

$0 - \csc^{2} (x) + 2 \csc (x) \cot (x)$

Step 2.4

Simplify.

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Step 2.4.1

Subtract $\csc^{2} (x)$ from $0$ .

$- \csc^{2} (x) + 2 \csc (x) \cot (x)$

Step 2.4.2

Reorder terms.

$2 \cot (x) \csc (x) - \csc^{2} (x)$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 4

Solve the original function $f (x) = 3 + \cot (x) - 2 \csc (x)$ at $x = \frac{π}{3}$ .

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Step 4.1

Replace the variable $x$ with $\frac{π}{3}$ in the expression.

$f (\frac{π}{3}) = 3 + \cot (\frac{π}{3}) - 2 \csc (\frac{π}{3})$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$f (\frac{π}{3}) = 3 + \frac{1}{\sqrt{3}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.2

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{π}{3}) = 3 + \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3

Combine and simplify the denominator.

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Step 4.2.1.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.4

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

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.5

Add $1$ and $1$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{{\sqrt{3}}^{2}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.6.2

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

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.6.3

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

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3^{\frac{2}{2}}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.3.6.4.1

Cancel the common factor.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3^{\frac{2}{2}}} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.6.4.2

Rewrite the expression.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \csc (\frac{π}{3})$

Step 4.2.1.3.6.5

Evaluate the exponent.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \csc (\frac{π}{3})$

Step 4.2.1.4

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2}{\sqrt{3}}$

Step 4.2.1.5

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.2.1.6

Combine and simplify the denominator.

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Step 4.2.1.6.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2.1.6.4

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

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.2.1.6.5

Add $1$ and $1$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.2.1.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.2.1.6.6.2

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

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.2.1.6.6.3

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

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.2.1.6.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.6.6.4.1

Cancel the common factor.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.2.1.6.6.4.2

Rewrite the expression.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.6.6.5

Evaluate the exponent.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - 2 \frac{2 \sqrt{3}}{3}$

Step 4.2.1.7

Multiply $- 2 \frac{2 \sqrt{3}}{3}$ .

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Step 4.2.1.7.1

Combine $- 2$ and $\frac{2 \sqrt{3}}{3}$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} + \frac{- 2 (2 \sqrt{3})}{3}$

Step 4.2.1.7.2

Multiply $2$ by $- 2$ .

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} + \frac{- 4 \sqrt{3}}{3}$

Step 4.2.1.8

Move the negative in front of the fraction.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3}}{3} - \frac{4 \sqrt{3}}{3}$

Step 4.2.2

Simplify terms.

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Step 4.2.2.1

Combine the numerators over the common denominator.

$f (\frac{π}{3}) = 3 + \frac{\sqrt{3} - 4 \sqrt{3}}{3}$

Step 4.2.2.2

Subtract $4 \sqrt{3}$ from $\sqrt{3}$ .

$f (\frac{π}{3}) = 3 + \frac{- 3 \sqrt{3}}{3}$

Step 4.2.2.3

Cancel the common factor of $- 3$ and $3$ .

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Step 4.2.2.3.1

Factor $3$ out of $- 3 \sqrt{3}$ .

$f (\frac{π}{3}) = 3 + \frac{3 (- \sqrt{3})}{3}$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $3$ out of $3$ .

$f (\frac{π}{3}) = 3 + \frac{3 (- \sqrt{3})}{3 (1)}$

Step 4.2.2.3.2.2

Cancel the common factor.

$f (\frac{π}{3}) = 3 + \frac{3 (- \sqrt{3})}{3 \cdot 1}$

Step 4.2.2.3.2.3

Rewrite the expression.

$f (\frac{π}{3}) = 3 + \frac{- \sqrt{3}}{1}$

Step 4.2.2.3.2.4

Divide $- \sqrt{3}$ by $1$ .

$f (\frac{π}{3}) = 3 - \sqrt{3}$

Step 4.2.3

The final answer is $3 - \sqrt{3}$ .

$3 - \sqrt{3}$

Step 5

Solve the original function $f (x) = 3 + \cot (x) - 2 \csc (x)$ at $x = \frac{5 π}{3}$ .

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Step 5.1

Replace the variable $x$ with $\frac{5 π}{3}$ in the expression.

$f (\frac{5 π}{3}) = 3 + \cot (\frac{5 π}{3}) - 2 \csc (\frac{5 π}{3})$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.

$f (\frac{5 π}{3}) = 3 - \cot (\frac{π}{3}) - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.2

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{1}{\sqrt{3}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{5 π}{3}) = 3 - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4

Combine and simplify the denominator.

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Step 5.2.1.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.4

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

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.5

Add $1$ and $1$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{{\sqrt{3}}^{2}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.6.2

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

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.6.3

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

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3^{\frac{2}{2}}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.1.4.6.4.1

Cancel the common factor.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3^{\frac{2}{2}}} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.6.4.2

Rewrite the expression.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.4.6.5

Evaluate the exponent.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 \csc (\frac{5 π}{3})$

Step 5.2.1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \csc (\frac{π}{3}))$

Step 5.2.1.6

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2}{\sqrt{3}})$

Step 5.2.1.7

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))$

Step 5.2.1.8

Combine and simplify the denominator.

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Step 5.2.1.8.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 5.2.1.8.2

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 5.2.1.8.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 5.2.1.8.4

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

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 5.2.1.8.5

Add $1$ and $1$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})$

Step 5.2.1.8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.1.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})$

Step 5.2.1.8.6.2

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

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Step 5.2.1.8.6.3

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

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 5.2.1.8.6.4

Cancel the common factor of $2$ .

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Step 5.2.1.8.6.4.1

Cancel the common factor.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 5.2.1.8.6.4.2

Rewrite the expression.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.8.6.5

Evaluate the exponent.

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} - 2 (- \frac{2 \sqrt{3}}{3})$

Step 5.2.1.9

Multiply $- 2 (- \frac{2 \sqrt{3}}{3})$ .

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Step 5.2.1.9.1

Multiply $- 1$ by $- 2$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} + 2 (\frac{2 \sqrt{3}}{3})$

Step 5.2.1.9.2

Combine $2$ and $\frac{2 \sqrt{3}}{3}$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} + \frac{2 (2 \sqrt{3})}{3}$

Step 5.2.1.9.3

Multiply $2$ by $2$ .

$f (\frac{5 π}{3}) = 3 - \frac{\sqrt{3}}{3} + \frac{4 \sqrt{3}}{3}$

Step 5.2.2

Simplify terms.

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Step 5.2.2.1

Combine the numerators over the common denominator.

$f (\frac{5 π}{3}) = 3 + \frac{- \sqrt{3} + 4 \sqrt{3}}{3}$

Step 5.2.2.2

Add $- \sqrt{3}$ and $4 \sqrt{3}$ .

$f (\frac{5 π}{3}) = 3 + \frac{3 \sqrt{3}}{3}$

Step 5.2.2.3

Cancel the common factor of $3$ .

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Step 5.2.2.3.1

Cancel the common factor.

$f (\frac{5 π}{3}) = 3 + \frac{3 \sqrt{3}}{3}$

Step 5.2.2.3.2

Divide $\sqrt{3}$ by $1$ .

$f (\frac{5 π}{3}) = 3 + \sqrt{3}$

Step 5.2.3

The final answer is $3 + \sqrt{3}$ .

$3 + \sqrt{3}$

Step 6

The horizontal tangent lines on function $f (x) = 3 + \cot (x) - 2 \csc (x)$ are $y = 3 - \sqrt{3}, y = 3 + \sqrt{3}$ .

$y = 3 - \sqrt{3}, y = 3 + \sqrt{3}$

Step 7