Trigonometry Examples

$\sec (45)$

Step 1

Convert the radian measure to degrees.

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Step 1.1

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

$(\sec (45)) \cdot \frac{180 °}{π}$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

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

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

Step 1.4.2

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} \cdot \frac{180}{π}$

Step 1.4.3

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} \cdot \frac{180}{π}$

Step 1.4.4

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} \cdot \frac{180}{π}$

Step 1.4.5

Add $1$ and $1$ .

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

Step 1.4.6

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

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Step 1.4.6.1

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

$\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} \cdot \frac{180}{π}$

Step 1.4.6.2

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

$\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} \cdot \frac{180}{π}$

Step 1.4.6.3

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

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

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

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

Step 1.4.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{2}}{2^{1}} \cdot \frac{180}{π}$

Step 1.4.6.5

Evaluate the exponent.

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

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $180$ .

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

$2 \sqrt{2} \cdot \frac{90}{π}$

Step 1.6

Combine $\frac{90}{π}$ and $2$ .

$\frac{90 \cdot 2}{π} \sqrt{2}$

Step 1.7

Multiply $90$ by $2$ .

$\frac{180}{π} \sqrt{2}$

Step 1.8

Combine $\frac{180}{π}$ and $\sqrt{2}$ .

$\frac{180 \sqrt{2}}{π}$

Step 1.9

$π$ is approximately equal to $3.14159265$ .

$\frac{180 \sqrt{2}}{3.14159265}$

Step 1.10

Convert to a decimal.

$81.02846845 °$

Step 2

The angle is in the first quadrant.

Quadrant $1$

Step 3