Trigonometry Examples

$\tan (22 + \tan (38 - \sqrt{3})) = k \tan (22 \tan (38))$

Step 1

Rewrite the equation as $k \tan (22 \tan (38)) = \tan (22 + \tan (38 - \sqrt{3}))$ .

$k \tan (22 \tan (38)) = \tan (22 + \tan (38 - \sqrt{3}))$

Step 2

Simplify.

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Step 2.1

Evaluate $\tan (38)$ .

$k \tan (22 \cdot 0.78128562) = \tan (22 + \tan (38 - \sqrt{3}))$

Step 2.2

Multiply $22$ by $0.78128562$ .

$k \tan (17.18828378) = \tan (22 + \tan (38 - \sqrt{3}))$

Step 3

Divide each term in $k \tan (17.18828378) = \tan (22 + \tan (38 - \sqrt{3}))$ by $\tan (17.18828378)$ and simplify.

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Step 3.1

Divide each term in $k \tan (17.18828378) = \tan (22 + \tan (38 - \sqrt{3}))$ by $\tan (17.18828378)$ .

$\frac{k \tan (17.18828378)}{\tan (17.18828378)} = \frac{\tan (22 + \tan (38 - \sqrt{3}))}{\tan (17.18828378)}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\tan (17.18828378)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{k \tan (17.18828378)}{\tan (17.18828378)} = \frac{\tan (22 + \tan (38 - \sqrt{3}))}{\tan (17.18828378)}$

Step 3.2.1.2

Divide $k$ by $1$ .

$k = \frac{\tan (22 + \tan (38 - \sqrt{3}))}{\tan (17.18828378)}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Rewrite $\tan (17.18828378)$ in terms of sines and cosines.

$k = \frac{\tan (22 + \tan (38 - \sqrt{3}))}{\frac{\sin (17.18828378)}{\cos (17.18828378)}}$

Step 3.3.2

Rewrite $\tan (22 + \tan (38 - \sqrt{3}))$ in terms of sines and cosines.

$k = \frac{\frac{\sin (22 + \tan (38 - \sqrt{3}))}{\cos (22 + \tan (38 - \sqrt{3}))}}{\frac{\sin (17.18828378)}{\cos (17.18828378)}}$

Step 3.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (17.18828378)}{\cos (17.18828378)}$ .

$k = \frac{\sin (22 + \tan (38 - \sqrt{3}))}{\cos (22 + \tan (38 - \sqrt{3}))} \cdot \frac{\cos (17.18828378)}{\sin (17.18828378)}$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Convert from $\frac{\sin (22 + \tan (38 - \sqrt{3}))}{\cos (22 + \tan (38 - \sqrt{3}))}$ to $\tan (22 + \tan (38 - \sqrt{3}))$ .

$k = \tan (22 + \tan (38 - \sqrt{3})) \frac{\cos (17.18828378)}{\sin (17.18828378)}$

Step 3.3.4.2

Convert from $\frac{\cos (17.18828378)}{\sin (17.18828378)}$ to $\cot (17.18828378)$ .

$k = \tan (22 + \tan (38 - \sqrt{3})) \cot (17.18828378)$

Step 3.3.5

Evaluate $\tan (38 - \sqrt{3})$ .

$k = \tan (22 + 0.73371214) \cot (17.18828378)$

Step 3.3.6

Add $22$ and $0.73371214$ .

$k = \tan (22.73371214) \cot (17.18828378)$

Step 4

The result can be shown in multiple forms.

Exact Form:

$k = \tan (22.73371214) \cot (17.18828378)$

Decimal Form:

$k = 1.35455264 \dots$