Precalculus Examples

$\sec (\frac{13 π}{12})$

Step 1

Replace $\sec (\frac{13 π}{12})$ with an equivalent expression $\frac{1}{\cos (\frac{13 π}{12})}$ using the fundamental identities.

$\frac{1}{\cos (\frac{13 π}{12})}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $\frac{13 π}{12}$ can be split into $\frac{3 π}{4} + \frac{π}{3}$ .

$\frac{1}{\cos (\frac{3 π}{4} + \frac{π}{3})}$

Step 2.2

Use the sum formula for cosine to simplify the expression. The formula states that $\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

$\frac{1}{\cos (\frac{π}{3}) \cdot \cos (\frac{3 π}{4}) - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.3

Remove parentheses.

$\frac{1}{\cos (\frac{π}{3}) \cdot \cos (\frac{3 π}{4}) - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.4

Simplify each term.

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Step 2.4.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{1}{\frac{1}{2} \cdot \cos (\frac{3 π}{4}) - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.4.2

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

$\frac{1}{\frac{1}{2} \cdot (- \cos (\frac{π}{4})) - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.4.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{1}{\frac{1}{2} \cdot (- \frac{\sqrt{2}}{2}) - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.4.4

Multiply $\frac{1}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 2.4.4.1

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

$\frac{1}{- \frac{\sqrt{2}}{2 \cdot 2} - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.4.4.2

Multiply $2$ by $2$ .

$\frac{1}{- \frac{\sqrt{2}}{4} - \sin (\frac{π}{3}) \cdot \sin (\frac{3 π}{4})}$

Step 2.4.5

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

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{3}}{2} \cdot \sin (\frac{3 π}{4})}$

Step 2.4.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{3}}{2} \cdot \sin (\frac{π}{4})}$

Step 2.4.7

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}$

Step 2.4.8

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

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Step 2.4.8.1

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

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2}}$

Step 2.4.8.2

Combine using the product rule for radicals.

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{2 \cdot 3}}{2 \cdot 2}}$

Step 2.4.8.3

Multiply $2$ by $3$ .

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{2 \cdot 2}}$

Step 2.4.8.4

Multiply $2$ by $2$ .

$\frac{1}{- \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}}$

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

$\frac{1}{\frac{- \sqrt{2} - \sqrt{6}}{4}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{4}{- \sqrt{2} - \sqrt{6}}$

Step 3.3

Multiply $\frac{4}{- \sqrt{2} - \sqrt{6}}$ by $1$ .

$\frac{4}{- \sqrt{2} - \sqrt{6}}$

Step 3.4

Multiply $\frac{4}{- \sqrt{2} - \sqrt{6}}$ by $\frac{- \sqrt{2} + \sqrt{6}}{- \sqrt{2} + \sqrt{6}}$ .

$\frac{4}{- \sqrt{2} - \sqrt{6}} \cdot \frac{- \sqrt{2} + \sqrt{6}}{- \sqrt{2} + \sqrt{6}}$

Step 3.5

Multiply $\frac{4}{- \sqrt{2} - \sqrt{6}}$ by $\frac{- \sqrt{2} + \sqrt{6}}{- \sqrt{2} + \sqrt{6}}$ .

$\frac{4 (- \sqrt{2} + \sqrt{6})}{(- \sqrt{2} - \sqrt{6}) (- \sqrt{2} + \sqrt{6})}$

Step 3.6

Expand the denominator using the FOIL method.

$\frac{4 (- \sqrt{2} + \sqrt{6})}{{\sqrt{2}}^{2} - \sqrt{12} + \sqrt{12} - {\sqrt{6}}^{2}}$

Step 3.7

Simplify.

$\frac{4 (- \sqrt{2} + \sqrt{6})}{- 4}$

Step 3.8

Simplify the expression.

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Step 3.8.1

Move the negative one from the denominator of $\frac{- \sqrt{2} + \sqrt{6}}{- 1}$ .

$- 1 \cdot (- \sqrt{2} + \sqrt{6})$

Step 3.8.2

Rewrite $- 1 \cdot (- \sqrt{2} + \sqrt{6})$ as $- (- \sqrt{2} + \sqrt{6})$ .

$- (- \sqrt{2} + \sqrt{6})$

Step 3.9

Apply the distributive property.

$- - \sqrt{2} - \sqrt{6}$

Step 3.10

Multiply $- - \sqrt{2}$ .

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Step 3.10.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{2} - \sqrt{6}$

Step 3.10.2

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

$\sqrt{2} - \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2} - \sqrt{6}$

Decimal Form:

$- 1.03527618 \dots$