Calculus Examples

$\arctan (\frac{2 + 3.6}{\frac{2}{\tan (30)} + \frac{3.6}{\tan (49.4)}})$

Step 1

Multiply the numerator and denominator of the complex fraction by $\tan (30) \tan (49.4)$ .

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Multiply $\frac{2 + 3.6}{\frac{2}{\tan (30)} + \frac{3.6}{\tan (49.4)}}$ by $\frac{\tan (30) \tan (49.4)}{\tan (30) \tan (49.4)}$ .

$\arctan (\frac{\tan (30) \tan (49.4)}{\tan (30) \tan (49.4)} \cdot \frac{2 + 3.6}{\frac{2}{\tan (30)} + \frac{3.6}{\tan (49.4)}})$

Combine.

$\arctan (\frac{\tan (30) \tan (49.4) (2 + 3.6)}{\tan (30) \tan (49.4) (\frac{2}{\tan (30)} + \frac{3.6}{\tan (49.4)})})$

Step 2

Apply the distributive property.

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (30) \tan (49.4) \frac{2}{\tan (30)} + \tan (30) \tan (49.4) \frac{3.6}{\tan (49.4)}})$

Step 3

Simplify by cancelling.

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Cancel the common factor of $\tan (30)$ .

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Factor $\tan (30)$ out of $\tan (30) \tan (49.4)$ .

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (30) (\tan (49.4)) \frac{2}{\tan (30)} + \tan (30) \tan (49.4) \frac{3.6}{\tan (49.4)}})$

Cancel the common factor.

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (30) \tan (49.4) \frac{2}{\tan (30)} + \tan (30) \tan (49.4) \frac{3.6}{\tan (49.4)}})$

Rewrite the expression.

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \frac{3.6}{\tan (49.4)}})$

Cancel the common factor of $\tan (49.4)$ .

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Factor $\tan (49.4)$ out of $\tan (30) \tan (49.4)$ .

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + \tan (49.4) \tan (30) \frac{3.6}{\tan (49.4)}})$

Cancel the common factor.

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + \tan (49.4) \tan (30) \frac{3.6}{\tan (49.4)}})$

Rewrite the expression.

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + \tan (30) \cdot 3.6})$

Multiply $\tan (30)$ by $3.6$ .

$\arctan (\frac{\tan (30) \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Step 4

Simplify the numerator.

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The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\arctan (\frac{\frac{\sqrt{3}}{3} \tan (49.4) \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Combine $\frac{\sqrt{3}}{3}$ and $\tan (49.4)$ .

$\arctan (\frac{\frac{\sqrt{3} \tan (49.4)}{3} \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Evaluate $\tan (49.4)$ .

$\arctan (\frac{\frac{\sqrt{3} \cdot 1.16672001}{3} \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Multiply $\sqrt{3}$ by $1.16672001$ .

$\arctan (\frac{\frac{2.02081835}{3} \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Divide $2.02081835$ by $3$ .

$\arctan (\frac{0.67360611 \cdot 2 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Multiply $0.67360611$ by $2$ .

$\arctan (\frac{1.34721223 + \tan (30) \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\arctan (\frac{1.34721223 + \frac{\sqrt{3}}{3} \tan (49.4) \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Combine $\frac{\sqrt{3}}{3}$ and $\tan (49.4)$ .

$\arctan (\frac{1.34721223 + \frac{\sqrt{3} \tan (49.4)}{3} \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Evaluate $\tan (49.4)$ .

$\arctan (\frac{1.34721223 + \frac{\sqrt{3} \cdot 1.16672001}{3} \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Multiply $\sqrt{3}$ by $1.16672001$ .

$\arctan (\frac{1.34721223 + \frac{2.02081835}{3} \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Divide $2.02081835$ by $3$ .

$\arctan (\frac{1.34721223 + 0.67360611 \cdot 3.6}{\tan (49.4) \cdot 2 + 2.07846096})$

Multiply $0.67360611$ by $3.6$ .

$\arctan (\frac{1.34721223 + 2.42498202}{\tan (49.4) \cdot 2 + 2.07846096})$

Add $1.34721223$ and $2.42498202$ .

$\arctan (\frac{3.77219425}{\tan (49.4) \cdot 2 + 2.07846096})$

Step 5

Simplify the denominator.

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Move $2$ to the left of $\tan (49.4)$ .

$\arctan (\frac{3.77219425}{2 \cdot \tan (49.4) + 2.07846096})$

Add $2 \tan (49.4)$ and $2.07846096$ .

$\arctan (\frac{3.77219425}{4.411901})$

Step 6

Divide $3.77219425$ by $4.411901$ .

$\arctan (0.85500428)$

Step 7

Evaluate $\arctan (0.85500428)$ .

$40.53058427$