Precalculus Examples

$\sec (45)$

Step 1

To convert degrees to radians, multiply by $\frac{π}{180 °}$ , since a full circle is $360 °$ or $2 π$ radians.

Step 2

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

$\frac{2}{\sqrt{2}} \cdot \frac{π}{180}$ radians

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Factor $2$ out of $180$ .

$\frac{2}{\sqrt{2}} \cdot \frac{π}{2 (90)}$ radians

Step 3.2

Cancel the common factor.

$\frac{2}{\sqrt{2}} \cdot \frac{π}{2 \cdot 90}$ radians

Step 3.3

Rewrite the expression.

$\frac{1}{\sqrt{2}} \cdot \frac{π}{90}$ radians

Step 4

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{π}{90}$ .

$\frac{π}{\sqrt{2} \cdot 90}$ radians

Step 5

Move $90$ to the left of $\sqrt{2}$ .

$\frac{π}{90 \sqrt{2}}$ radians

Step 6

Multiply $\frac{π}{90 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{π}{90 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$ radians

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{π}{90 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{π \sqrt{2}}{90 \sqrt{2} \sqrt{2}}$ radians

Step 7.2

Move $\sqrt{2}$ .

$\frac{π \sqrt{2}}{90 (\sqrt{2} \sqrt{2})}$ radians

Step 7.3

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

$\frac{π \sqrt{2}}{90 (\sqrt{2} \sqrt{2})}$ radians

Step 7.4

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

$\frac{π \sqrt{2}}{90 (\sqrt{2} \sqrt{2})}$ radians

Step 7.5

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

$\frac{π \sqrt{2}}{90 {\sqrt{2}}^{1 + 1}}$ radians

Step 7.6

Add $1$ and $1$ .

$\frac{π \sqrt{2}}{90 {\sqrt{2}}^{2}}$ radians

Step 7.7

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

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Step 7.7.1

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

$\frac{π \sqrt{2}}{90 {(2^{\frac{1}{2}})}^{2}}$ radians

Step 7.7.2

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

$\frac{π \sqrt{2}}{90 \cdot 2^{\frac{1}{2} \cdot 2}}$ radians

Step 7.7.3

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

$\frac{π \sqrt{2}}{90 \cdot 2^{\frac{2}{2}}}$ radians

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

$\frac{π \sqrt{2}}{90 \cdot 2^{\frac{2}{2}}}$ radians

Step 7.7.4.2

Rewrite the expression.

$\frac{π \sqrt{2}}{90 \cdot 2}$ radians

Step 7.7.5

Evaluate the exponent.

$\frac{π \sqrt{2}}{90 \cdot 2}$ radians

Step 8

Multiply $90$ by $2$ .

$\frac{π \sqrt{2}}{180}$ radians