Precalculus Examples

$\frac{\sin (60) \cdot 200}{\sin (45)} \cdot \log (7)$

Step 1

Separate fractions.

$\frac{200}{1} \cdot \frac{\sin (60)}{\sin (45)} \cdot \log (7)$

Step 2

Rewrite $\frac{\sin (60)}{\sin (45)}$ as a product.

$\frac{200}{1} (\sin (60) \frac{1}{\sin (45)}) \cdot \log (7)$

Step 3

Write $\sin (60)$ as a fraction with denominator $1$ .

$\frac{200}{1} (\frac{\sin (60)}{1} \cdot \frac{1}{\sin (45)}) \cdot \log (7)$

Step 4

Simplify.

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Step 4.1

Divide $\sin (60)$ by $1$ .

$\frac{200}{1} (\sin (60) \frac{1}{\sin (45)}) \cdot \log (7)$

Step 4.2

Convert from $\frac{1}{\sin (45)}$ to $\csc (45)$ .

$\frac{200}{1} (\sin (60) \csc (45)) \cdot \log (7)$

Step 5

Divide $200$ by $1$ .

$200 (\sin (60) \csc (45)) \cdot \log (7)$

Step 6

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$200 (\frac{\sqrt{3}}{2} \csc (45)) \cdot \log (7)$

Step 7

The exact value of $\csc (45)$ is $\sqrt{2}$ .

$200 (\frac{\sqrt{3}}{2} \sqrt{2}) \cdot \log (7)$

Step 8

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

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Step 8.1

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

$200 \frac{\sqrt{3} \sqrt{2}}{2} \cdot \log (7)$

Step 8.2

Combine using the product rule for radicals.

$200 \frac{\sqrt{3 \cdot 2}}{2} \cdot \log (7)$

Step 8.3

Multiply $3$ by $2$ .

$200 \frac{\sqrt{6}}{2} \cdot \log (7)$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Factor $2$ out of $200$ .

$2 (100) \frac{\sqrt{6}}{2} \cdot \log (7)$

Step 9.2

Cancel the common factor.

$2 \cdot 100 \frac{\sqrt{6}}{2} \cdot \log (7)$

Step 9.3

Rewrite the expression.

$100 \sqrt{6} \cdot \log (7)$

Step 10

Multiply $100 \sqrt{6} \log (7)$ .

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Step 10.1

Reorder $\log (7)$ and $100$ .

$100 \cdot \log (7) \sqrt{6}$

Step 10.2

Simplify $100 \cdot \log (7)$ by moving $100$ inside the logarithm.

$\log (7^{100}) \sqrt{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\log (7^{100}) \sqrt{6}$

Decimal Form:

$207.00589806 \dots$