Use a calculator to evaluate each of the following (a) \( \cos (A-B) \) (b) \( \cos A-\cos B \) (c) \( \cos A \cdot \cos B+\sin A \cdot \sin B \)

To evaluate the expressions you provided, you'll want to use known values for angles \( A \) and \( B \). Let's say \( A = 30^\circ \) and \( B = 45^\circ \) as sample values: (a) For \( \cos (A - B) \): Using the difference of angles formula, \( \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \). Calculating this: - \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \) - \( \sin(30^\circ) = \frac{1}{2} \) - \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \) - \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \) So, \[ \cos(30^\circ - 45^\circ) = \cos(-15^\circ) = \cos(15^\circ) \approx 0.9659 \] (b) For \( \cos A - \cos B \): Calculating this directly: \[ \cos(30^\circ) - \cos(45^\circ) = \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \approx 0.3660 - 0.7071 \approx -0.3411 \] (c) For \( \cos A \cdot \cos B + \sin A \cdot \sin B \): Using the values calculated earlier: \[ \cos(30^\circ) \cdot \cos(45^\circ) + \sin(30^\circ) \cdot \sin(45^\circ) = \left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}\right) + \left(\frac{1}{2} \cdot \frac{\sqrt{2}}{2}\right) \] Calculating it gives: \[ \approx 0.6124 + 0.3536 \approx 0.9660 \] In summary: - \( \cos(A - B) \approx 0.9659 \) - \( \cos A - \cos B \approx -0.3411 \) - \( \cos A \cdot \cos B + \sin A \cdot \sin B \approx 0.9660 \)