Cosine 105 A8877D

1. **Problem:** Find the value of $\cos 105^\circ$ without using a calculator and express it in the form $\frac{1-\sqrt{a}}{b\sqrt{b}}$. 2. **Formula:** Use the cosine addition formula: $$\cos (x + y) = \cos x \cos y - \sin x \sin y$$ 3. **Apply formula:** Since $105^\circ = 60^\circ + 45^\circ$, we have: $$\cos 105^\circ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ$$ 4. **Known values:** $$\cos 60^\circ = \frac{1}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}$$ 5. **Substitute:** $$\cos 105^\circ = \frac{1}{2} \times \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$$ 6. **Combine terms:** $$\cos 105^\circ = \frac{\sqrt{2} - \sqrt{6}}{4}$$ 7. **Rewrite in requested form:** Multiply numerator and denominator by $\sqrt{2}$ to rationalize denominator: $$\cos 105^\circ = \frac{\sqrt{2} - \sqrt{6}}{4} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2 - \sqrt{12}}{4 \sqrt{2}}$$ 8. **Simplify $\sqrt{12}$:** $$\sqrt{12} = 2\sqrt{3}$$ 9. **Final expression:** $$\cos 105^\circ = \frac{2 - 2\sqrt{3}}{4 \sqrt{2}} = \frac{2(1 - \sqrt{3})}{4 \sqrt{2}} = \frac{1 - \sqrt{3}}{2 \sqrt{2}}$$ This matches the form $\frac{1 - \sqrt{a}}{b \sqrt{b}}$ with $a=3$ and $b=2$. **Answer:** $$\boxed{\cos 105^\circ = \frac{1 - \sqrt{3}}{2 \sqrt{2}}}$$