Using the sum formula for cosine, express \( \cos(a + b) \) in terms of \( \cos a, \cos b, \sin a, \text{ and } \sin b \).

The sum formula for cosine allows you to express \( \cos(a + b) \) in terms of the cosines and sines of the individual angles \( a \) and \( b \). The formula is derived from the addition of angles in trigonometry and is given by: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] **Explanation:** 1. **Cosine of a Sum:** The cosine of the sum of two angles \( a \) and \( b \) can be expanded using the sum formula: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \] 2. **Components:** - \( \cos a \cos b \): This term represents the product of the cosines of the individual angles. - \( \sin a \sin b \): This term represents the product of the sines of the individual angles. - The minus sign between the two terms indicates that the sine product is subtracted from the cosine product. **Example Usage:** If you want to calculate \( \cos(30^\circ + 45^\circ) \), you can apply the formula as follows: \[ \cos(75^\circ) = \cos 30^\circ \cos 45^\circ - \sin 30^\circ \sin 45^\circ \] Using known values: \[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 30^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2} \] Plugging these in: \[ \cos(75^\circ) = \left( \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} \right) - \left( \frac{1}{2} \times \frac{\sqrt{2}}{2} \right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \] This formula is fundamental in trigonometry and is widely used in various applications, including solving triangles, oscillatory motion, and wave analysis. **Summary Formula:** \[ \boxed{\cos(a + b) = \cos a\, \cos b\; -\; \sin a\, \sin b} \]