Calculating Cosine 105 Degrees

To find the value of \(\cos(105^\circ)\), we break down \(105^\circ\) into the sum of two standard angles: \(105^\circ = 60^\circ + 45^\circ\). This allows us to use trigonometric identities.

Applying the Cosine Sum Identity

The angle addition identity for cosine states: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)

Applying this identity with \(A = 60^\circ\) and \(B = 45^\circ\): \(\cos(105^\circ) = \cos(60^\circ + 45^\circ) = \cos(60^\circ)\cos(45^\circ) - \sin(60^\circ)\sin(45^\circ)\)

Substituting Known Trigonometric Values

We use the known values for these standard angles:

• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
• \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
• \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)

Performing the Calculation

Substitute these values into the expanded formula:

\(\cos(105^\circ) = \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right)\)

Simplify the terms:

\(\cos(105^\circ) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\)

Combine the terms over the common denominator:

\(\cos(105^\circ) = \frac{\sqrt{2} - \sqrt{6}}{4}\)

Matching the Result Format

The calculated value is \(\frac{\sqrt{2} - \sqrt{6}}{4}\). Let's verify Option A: \(\frac{(1-\sqrt{3})}{2\sqrt{2}}\). To compare, we rationalize the denominator of Option A:

\(\frac{(1-\sqrt{3})}{2\sqrt{2}} = \frac{(1-\sqrt{3}) \times \sqrt{2}}{(2\sqrt{2}) \times \sqrt{2}} = \frac{\sqrt{2} - \sqrt{6}}{2 \times 2} = \frac{\sqrt{2} - \sqrt{6}}{4}\)

The calculated value matches Option A.