Trigonometry Examples

$\cos (- 105)$

Step 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 $45 - 150$ .

$\cos (45 - 150)$

Step 2

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

$\cos (45) \cdot \cos (150) + \sin (45) \cdot \sin (150)$

Step 3

Remove parentheses.

$\cos (45) \cdot \cos (150) + \sin (45) \cdot \sin (150)$

Step 4

Simplify each term.

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Step 4.1

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

$\frac{\sqrt{2}}{2} \cdot \cos (150) + \sin (45) \cdot \sin (150)$

Step 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.

$\frac{\sqrt{2}}{2} \cdot (- \cos (30)) + \sin (45) \cdot \sin (150)$

Step 4.3

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

$\frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{3}}{2}) + \sin (45) \cdot \sin (150)$

Step 4.4

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

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Step 4.4.1

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

$- \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \sin (45) \cdot \sin (150)$

Step 4.4.2

Combine using the product rule for radicals.

$- \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \sin (45) \cdot \sin (150)$

Step 4.4.3

Multiply $2$ by $3$ .

$- \frac{\sqrt{6}}{2 \cdot 2} + \sin (45) \cdot \sin (150)$

Step 4.4.4

Multiply $2$ by $2$ .

$- \frac{\sqrt{6}}{4} + \sin (45) \cdot \sin (150)$

Step 4.5

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

$- \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \sin (150)$

Step 4.6

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

$- \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \sin (30)$

Step 4.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

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

Step 4.8

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

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Step 4.8.1

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

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

Step 4.8.2

Multiply $2$ by $2$ .

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

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Factor $- 1$ out of $- \sqrt{6}$ .

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

Step 5.3

Factor $- 1$ out of $\sqrt{2}$ .

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

Step 5.4

Factor $- 1$ out of $- (\sqrt{6}) - 1 (- \sqrt{2})$ .

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

Step 5.5

Simplify the expression.

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Step 5.5.1

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

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

Step 5.5.2

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$