Trigonometry Examples

$\sec (105)$

Step 1

Replace $\sec (105)$ with an equivalent expression $\frac{1}{\cos (105)}$ using the fundamental identities.

$\frac{1}{\cos (105)}$

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, $105$ can be split into $45 + 60$ .

$\frac{1}{\cos (45 + 60)}$

Step 2.2

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

$\frac{1}{\cos (60) \cdot \cos (45) - \sin (60) \cdot \sin (45)}$

Step 2.3

Remove parentheses.

$\frac{1}{\cos (60) \cdot \cos (45) - \sin (60) \cdot \sin (45)}$

Step 2.4

Simplify each term.

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Step 2.4.1

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\frac{1}{\frac{1}{2} \cdot \cos (45) - \sin (60) \cdot \sin (45)}$

Step 2.4.2

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

$\frac{1}{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \sin (60) \cdot \sin (45)}$

Step 2.4.3

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

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Step 2.4.3.1

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

$\frac{1}{\frac{\sqrt{2}}{2 \cdot 2} - \sin (60) \cdot \sin (45)}$

Step 2.4.3.2

Multiply $2$ by $2$ .

$\frac{1}{\frac{\sqrt{2}}{4} - \sin (60) \cdot \sin (45)}$

Step 2.4.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\frac{\sqrt{2}}{4} - \frac{\sqrt{3}}{2} \cdot \sin (45)}$

Step 2.4.5

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

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

Step 2.4.6

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

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Step 2.4.6.1

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 2.4.6.2

Combine using the product rule for radicals.

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

Step 2.4.6.3

Multiply $2$ by $3$ .

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

Step 2.4.6.4

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 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 3.86370330 \dots$