Question

Value of $ \cos 105^\circ $

• A. \( \frac{\sqrt{3} + 1}{2\sqrt{2}} \)
• B. \( \frac{\sqrt{3} - 1}{2\sqrt{2}} \)
• C. \( \frac{\sqrt{3} + 1}{2\sqrt{2}} \)
• D. \( \frac{1 - \sqrt{3}}{2\sqrt{2}} \)

Hint:

For angles like \( 105^\circ \), break them into known angles (e.g., \( 180^\circ - 75^\circ \)) and use the angle addition formula for cosine to simplify the expression.

Answer 1

The Correct Option is B

We know that: \[ \cos(180^\circ - 75^\circ) = -\cos 75^\circ \] Using the formula for \( \cos(A + B) \), where \( A = 45^\circ \) and \( B = 30^\circ \): \[ \cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \] Substitute the known values of \( \cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}} \), \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), and \( \sin 30^\circ = \frac{1}{2} \): \[ \cos(75^\circ) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \] Thus: \[ \cos 105^\circ = -\cos 75^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]