Precalculus Examples

$\sec (210)$

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.

$(\sec (210)) \cdot \frac{180 °}{π}$

Step 1.2

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

$- \sec (30) \cdot \frac{180}{π}$

Step 1.3

The exact value of $\sec (30)$ is $\frac{2}{\sqrt{3}}$ .

$- \frac{2}{\sqrt{3}} \cdot \frac{180}{π}$

Step 1.4

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

$- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) \cdot \frac{180}{π}$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

$- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot \frac{180}{π}$

Step 1.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot \frac{180}{π}$

Step 1.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot \frac{180}{π}$

Step 1.5.4

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

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot \frac{180}{π}$

Step 1.5.5

Add $1$ and $1$ .

$- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \cdot \frac{180}{π}$

Step 1.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot \frac{180}{π}$

Step 1.5.6.2

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

$- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot \frac{180}{π}$

Step 1.5.6.3

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

$- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot \frac{180}{π}$

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.

$- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \cdot \frac{180}{π}$

Step 1.5.6.4.2

Rewrite the expression.

$- \frac{2 \sqrt{3}}{3^{1}} \cdot \frac{180}{π}$

Step 1.5.6.5

Evaluate the exponent.

$- \frac{2 \sqrt{3}}{3} \cdot \frac{180}{π}$

Step 1.6

Cancel the common factor of $3$ .

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Step 1.6.1

Move the leading negative in $- \frac{2 \sqrt{3}}{3}$ into the numerator.

$\frac{- 2 \sqrt{3}}{3} \cdot \frac{180}{π}$

Step 1.6.2

Factor $3$ out of $180$ .

$\frac{- 2 \sqrt{3}}{3} \cdot \frac{3 (60)}{π}$

Step 1.6.3

Cancel the common factor.

$\frac{- 2 \sqrt{3}}{3} \cdot \frac{3 \cdot 60}{π}$

Step 1.6.4

Rewrite the expression.

$- 2 \sqrt{3} \cdot \frac{60}{π}$

Step 1.7

Combine $\frac{60}{π}$ and $- 2$ .

$\frac{60 \cdot - 2}{π} \sqrt{3}$

Step 1.8

Multiply $60$ by $- 2$ .

$\frac{- 120}{π} \sqrt{3}$

Step 1.9

Combine $\frac{- 120}{π}$ and $\sqrt{3}$ .

$\frac{- 120 \sqrt{3}}{π}$

Step 1.10

Move the negative in front of the fraction.

$- \frac{120 \sqrt{3}}{π}$

Step 1.11

$π$ is approximately equal to $3.14159265$ .

$- \frac{120 \sqrt{3}}{3.14159265}$

Step 1.12

Convert to a decimal.

$- 66.15946745 °$

Step 2

For angles smaller than $0 °$ , add $360 °$ to the angle until the angle is larger than $0 °$ .

$293.84053254$

Step 3

The angle is in the fourth quadrant.

Quadrant $4$

Step 4