Calculus Examples

$y = \frac{8}{\cos (\frac{1}{\sqrt{2}})} - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1

Simplify each term.

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Step 1.1

Separate fractions.

$y = \frac{8}{1} \cdot \frac{1}{\cos (\frac{1}{\sqrt{2}})} - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.2

Convert from $\frac{1}{\cos (\frac{1}{\sqrt{2}})}$ to $\sec (\frac{1}{\sqrt{2}})$ .

$y = \frac{8}{1} \sec (\frac{1}{\sqrt{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.3

Divide $8$ by $1$ .

$y = 8 \sec (\frac{1}{\sqrt{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.4

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

$y = 8 \sec (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$y = 8 \sec (\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = 8 \sec (\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = 8 \sec (\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.4

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

$y = 8 \sec (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.5

Add $1$ and $1$ .

$y = 8 \sec (\frac{\sqrt{2}}{{\sqrt{2}}^{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.6

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

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Step 1.5.6.1

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

$y = 8 \sec (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.6.2

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

$y = 8 \sec (\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.6.3

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

$y = 8 \sec (\frac{\sqrt{2}}{2^{\frac{2}{2}}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$y = 8 \sec (\frac{\sqrt{2}}{2^{\frac{2}{2}}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.6.4.2

Rewrite the expression.

$y = 8 \sec (\frac{\sqrt{2}}{2^{1}}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.5.6.5

Evaluate the exponent.

$y = 8 \sec (\frac{\sqrt{2}}{2}) - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.6

Evaluate $\sec (\frac{\sqrt{2}}{2})$ .

$y = 8 \cdot 1.31536613 - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.7

Multiply $8$ by $1.31536613$ .

$y = 10.5229291 - \frac{5}{\sin (\frac{1}{\sqrt{2}})}$

Step 1.8

Separate fractions.

$y = 10.5229291 - (\frac{5}{1} \cdot \frac{1}{\sin (\frac{1}{\sqrt{2}})})$

Step 1.9

Convert from $\frac{1}{\sin (\frac{1}{\sqrt{2}})}$ to $\csc (\frac{1}{\sqrt{2}})$ .

$y = 10.5229291 - (\frac{5}{1} \csc (\frac{1}{\sqrt{2}}))$

Step 1.10

Divide $5$ by $1$ .

$y = 10.5229291 - (5 \csc (\frac{1}{\sqrt{2}}))$

Step 1.11

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

$y = 10.5229291 - (5 \csc (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))$

Step 1.12

Combine and simplify the denominator.

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Step 1.12.1

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

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}))$

Step 1.12.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}))$

Step 1.12.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}))$

Step 1.12.4

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

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}))$

Step 1.12.5

Add $1$ and $1$ .

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{{\sqrt{2}}^{2}}))$

Step 1.12.6

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

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Step 1.12.6.1

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

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}))$

Step 1.12.6.2

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

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}))$

Step 1.12.6.3

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

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{2^{\frac{2}{2}}}))$

Step 1.12.6.4

Cancel the common factor of $2$ .

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Step 1.12.6.4.1

Cancel the common factor.

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{2^{\frac{2}{2}}}))$

Step 1.12.6.4.2

Rewrite the expression.

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{2^{1}}))$

Step 1.12.6.5

Evaluate the exponent.

$y = 10.5229291 - (5 \csc (\frac{\sqrt{2}}{2}))$

Step 1.13

Evaluate $\csc (\frac{\sqrt{2}}{2})$ .

$y = 10.5229291 - (5 \cdot 1.53932133)$

Step 1.14

Multiply $- (5 \cdot 1.53932133)$ .

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Step 1.14.1

Multiply $5$ by $1.53932133$ .

$y = 10.5229291 - 1 \cdot 7.69660667$

Step 1.14.2

Multiply $- 1$ by $7.69660667$ .

$y = 10.5229291 - 7.69660667$

Step 2

Subtract $7.69660667$ from $10.5229291$ .

$y = 2.82632243$