Expand Using Sum/Difference Formulas cos(105 degrees )

$\cos (105 °)$

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 $45 + 60$ .

$\cos (45 + 60)$

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

$\cos (60) \cdot \cos (45) - \sin (60) \cdot \sin (45)$

Remove parentheses.

$\cos (60) \cdot \cos (45) - \sin (60) \cdot \sin (45)$

Simplify each term.

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$

Expand Using Sum/Difference Formulas cos(105 degrees )

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