Trigonometry Examples

$\sec (- \frac{7 π}{12})$

Step 1

Replace $\sec (- \frac{7 π}{12})$ with an equivalent expression $\frac{1}{\cos (- \frac{7 π}{12})}$ using the fundamental identities.

$\frac{1}{\cos (- \frac{7 π}{12})}$

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, $- \frac{7 π}{12}$ can be split into $\frac{π}{4} - \frac{5 π}{6}$ .

$\frac{1}{\cos (\frac{π}{4} - \frac{5 π}{6})}$

Step 2.2

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

$\frac{1}{\cos (\frac{π}{4}) \cdot \cos (\frac{5 π}{6}) + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.3

Remove parentheses.

$\frac{1}{\cos (\frac{π}{4}) \cdot \cos (\frac{5 π}{6}) + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4

Simplify each term.

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Step 2.4.1

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

$\frac{1}{\frac{\sqrt{2}}{2} \cdot \cos (\frac{5 π}{6}) + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

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.

$\frac{1}{\frac{\sqrt{2}}{2} \cdot (- \cos (\frac{π}{6})) + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{3}}{2}) + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4.4

Multiply $\frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2})$ .

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Step 2.4.4.1

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

$\frac{1}{- \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4.4.2

Combine using the product rule for radicals.

$\frac{1}{- \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4.4.3

Multiply $2$ by $3$ .

$\frac{1}{- \frac{\sqrt{6}}{2 \cdot 2} + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4.4.4

Multiply $2$ by $2$ .

$\frac{1}{- \frac{\sqrt{6}}{4} + \sin (\frac{π}{4}) \cdot \sin (\frac{5 π}{6})}$

Step 2.4.5

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

$\frac{1}{- \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \sin (\frac{5 π}{6})}$

Step 2.4.6

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

$\frac{1}{- \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \sin (\frac{π}{6})}$

Step 2.4.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 2.4.8

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

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Step 2.4.8.1

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

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

Step 2.4.8.2

Multiply $2$ by $2$ .

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

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Expand the denominator using the FOIL method.

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

Step 3.7

Simplify.

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

Step 3.8

Cancel the common factor of $4$ .

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Step 3.8.1

Cancel the common factor.

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

Step 3.8.2

Divide $- \sqrt{6} - \sqrt{2}$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 3.86370330 \dots$