Precalculus Examples

$\cos^{3} (\frac{5 π}{6}) \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1

Simplify each term.

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Step 1.1

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

${(- \cos (\frac{π}{6}))}^{3} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

${(- \frac{\sqrt{3}}{2})}^{3} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.3

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

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Step 1.3.1

Apply the product rule to $- \frac{\sqrt{3}}{2}$ .

${(- 1)}^{3} {(\frac{\sqrt{3}}{2})}^{3} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.3.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

${(- 1)}^{3} \frac{{\sqrt{3}}^{3}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.4

Raise $- 1$ to the power of $3$ .

$- \frac{{\sqrt{3}}^{3}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

$- \frac{\sqrt{3^{3}}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.5.2

Raise $3$ to the power of $3$ .

$- \frac{\sqrt{27}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.5.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.5.3.1

Factor $9$ out of $27$ .

$- \frac{\sqrt{9 (3)}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.5.3.2

Rewrite $9$ as $3^{2}$ .

$- \frac{\sqrt{3^{2} \cdot 3}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.5.4

Pull terms out from under the radical.

$- \frac{3 \sqrt{3}}{2^{3}} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.6

Raise $2$ to the power of $3$ .

$- \frac{3 \sqrt{3}}{8} \cdot \sin^{2} (\frac{7 π}{8}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7

The exact value of $\sin (\frac{7 π}{8})$ is $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.7.1

Rewrite $\frac{7 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$- \frac{3 \sqrt{3}}{8} \cdot \sin^{2} (\frac{\frac{7 π}{4}}{2}) + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.2

Apply the sine half-angle identity.

$- \frac{3 \sqrt{3}}{8} \cdot {(\pm \sqrt{\frac{1 - \cos (\frac{7 π}{4})}{2}})}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.3

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{1 - \cos (\frac{7 π}{4})}{2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4

Simplify $\sqrt{\frac{1 - \cos (\frac{7 π}{4})}{2}}$ .

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Step 1.7.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.3

Write $1$ as a fraction with a common denominator.

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.4

Combine the numerators over the common denominator.

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.5

Multiply the numerator by the reciprocal of the denominator.

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.6

Multiply $\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.7.4.6.1

Multiply $\frac{2 - \sqrt{2}}{2}$ by $\frac{1}{2}$ .

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.6.2

Multiply $2$ by $2$ .

$- \frac{3 \sqrt{3}}{8} \cdot {\sqrt{\frac{2 - \sqrt{2}}{4}}}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$- \frac{3 \sqrt{3}}{8} \cdot {(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}})}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.8

Simplify the denominator.

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Step 1.7.4.8.1

Rewrite $4$ as $2^{2}$ .

$- \frac{3 \sqrt{3}}{8} \cdot {(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}})}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.7.4.8.2

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{3 \sqrt{3}}{8} \cdot {(\frac{\sqrt{2 - \sqrt{2}}}{2})}^{2} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.8

Apply the product rule to $\frac{\sqrt{2 - \sqrt{2}}}{2}$ .

$- \frac{3 \sqrt{3}}{8} \cdot \frac{{\sqrt{2 - \sqrt{2}}}^{2}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.9

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

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Step 1.9.1

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

$- \frac{3 \sqrt{3}}{8} \cdot \frac{{({(2 - \sqrt{2})}^{\frac{1}{2}})}^{2}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.9.2

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

$- \frac{3 \sqrt{3}}{8} \cdot \frac{{(2 - \sqrt{2})}^{\frac{1}{2} \cdot 2}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.9.3

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

$- \frac{3 \sqrt{3}}{8} \cdot \frac{{(2 - \sqrt{2})}^{\frac{2}{2}}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

$- \frac{3 \sqrt{3}}{8} \cdot \frac{{(2 - \sqrt{2})}^{\frac{2}{2}}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.9.4.2

Rewrite the expression.

$- \frac{3 \sqrt{3}}{8} \cdot \frac{{(2 - \sqrt{2})}^{1}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.9.5

Simplify.

$- \frac{3 \sqrt{3}}{8} \cdot \frac{2 - \sqrt{2}}{2^{2}} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.10

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

$- \frac{3 \sqrt{3}}{8} \cdot \frac{2 - \sqrt{2}}{4} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.11

Multiply $- \frac{3 \sqrt{3}}{8} \cdot \frac{2 - \sqrt{2}}{4}$ .

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Step 1.11.1

Multiply $\frac{2 - \sqrt{2}}{4}$ by $\frac{3 \sqrt{3}}{8}$ .

$- \frac{(2 - \sqrt{2}) (3 \sqrt{3})}{4 \cdot 8} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.11.2

Multiply $4$ by $8$ .

$- \frac{(2 - \sqrt{2}) (3 \sqrt{3})}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.12

Group $\sqrt{3}$ and $2 - \sqrt{2}$ together.

$- \frac{\sqrt{3} (2 - \sqrt{2}) \cdot 3}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.13

Apply the distributive property.

$- \frac{(\sqrt{3} \cdot 2 + \sqrt{3} (- \sqrt{2})) \cdot 3}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.14

Move $2$ to the left of $\sqrt{3}$ .

$- \frac{(2 \cdot \sqrt{3} + \sqrt{3} (- \sqrt{2})) \cdot 3}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.15

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

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Step 1.15.1

Combine using the product rule for radicals.

$- \frac{(2 \cdot \sqrt{3} - \sqrt{3 \cdot 2}) \cdot 3}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.15.2

Multiply $3$ by $2$ .

$- \frac{(2 \cdot \sqrt{3} - \sqrt{6}) \cdot 3}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

$- \frac{(2 \sqrt{3} - \sqrt{6}) \cdot 3}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.16

Move $3$ to the left of $2 \sqrt{3} - \sqrt{6}$ .

$- \frac{3 \cdot (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{π}{6} \cdot \ln (8))}{\sqrt{7}}$

Step 1.17

Simplify the numerator.

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Step 1.17.1

Combine $\frac{π}{6}$ and $\ln (8)$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{π \ln (8)}{6})}{\sqrt{7}}$

Step 1.17.2

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{π \ln (2^{3})}{6})}{\sqrt{7}}$

Step 1.17.3

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{π (3 \ln (2))}{6})}{\sqrt{7}}$

Step 1.17.4

Cancel the common factor of $3$ and $6$ .

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Step 1.17.4.1

Factor $3$ out of $π (3 \ln (2))$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{3 (π (\ln (2)))}{6})}{\sqrt{7}}$

Step 1.17.4.2

Cancel the common factors.

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Step 1.17.4.2.1

Factor $3$ out of $6$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{3 (π (\ln (2)))}{3 \cdot 2})}{\sqrt{7}}$

Step 1.17.4.2.2

Cancel the common factor.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{3 (π (\ln (2)))}{3 \cdot 2})}{\sqrt{7}}$

Step 1.17.4.2.3

Rewrite the expression.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\tan (\frac{π (\ln (2))}{2})}{\sqrt{7}}$

Step 1.17.5

Rewrite $\tan (\frac{π \ln (2)}{2})$ in terms of sines and cosines.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})}}{\sqrt{7}}$

Step 1.18

Multiply the numerator by the reciprocal of the denominator.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{1}{\sqrt{7}}$

Step 1.19

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} (\frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 1.20

Combine and simplify the denominator.

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Step 1.20.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 1.20.2

Raise $\sqrt{7}$ to the power of $1$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 1.20.3

Raise $\sqrt{7}$ to the power of $1$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 1.20.4

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

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 1.20.5

Add $1$ and $1$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Step 1.20.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 1.20.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 1.20.6.2

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

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 1.20.6.3

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

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.20.6.4

Cancel the common factor of $2$ .

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Step 1.20.6.4.1

Cancel the common factor.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.20.6.4.2

Rewrite the expression.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{7^{1}}$

Step 1.20.6.5

Evaluate the exponent.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})} \cdot \frac{\sqrt{7}}{7}$

Step 1.21

Multiply $\frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})}$ by $\frac{\sqrt{7}}{7}$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2}) \sqrt{7}}{\cos (\frac{π \ln (2)}{2}) \cdot 7}$

Step 1.22

Move $7$ to the left of $\cos (\frac{π \ln (2)}{2})$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sin (\frac{π \ln (2)}{2}) \sqrt{7}}{7 \cos (\frac{π \ln (2)}{2})}$

Step 2

Simplify each term.

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Step 2.1

Separate fractions.

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sqrt{7}}{7} \cdot \frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})}$

Step 2.2

Convert from $\frac{\sin (\frac{π \ln (2)}{2})}{\cos (\frac{π \ln (2)}{2})}$ to $\tan (\frac{π \ln (2)}{2})$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sqrt{7}}{7} \tan (\frac{π \ln (2)}{2})$

Step 2.3

Combine $\frac{\sqrt{7}}{7}$ and $\tan (\frac{π \ln (2)}{2})$ .

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sqrt{7} \tan (\frac{π \ln (2)}{2})}{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 (2 \sqrt{3} - \sqrt{6})}{32} + \frac{\sqrt{7} \tan (\frac{π \ln (2)}{2})}{7}$

Decimal Form:

$0.62734487 \dots$