Given: if \( A=69 \) and \( B=30 \) (ii) \( A=225 \) and \( B=135 \) Eire a caiculator in evahums canthof the following: (a) \( \cos (A-B) \) (b) \( \cos A \cos B \) (c) \( \cos A \cdot \cos B+\sin A \sin B \)
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To solve for the cosine values using a calculator, let's first clarify the values given in degrees: - For the first case: \( A = 69^\circ \) and \( B = 30^\circ \). - For the second case: \( A = 225^\circ \) and \( B = 135^\circ \). Now, let’s calculate: (a) For \( A = 69^\circ \) and \( B = 30^\circ \): \[ \cos(A-B) = \cos(69^\circ - 30^\circ) = \cos(39^\circ) \approx 0.7771 \] For \( A = 225^\circ \) and \( B = 135^\circ \): \[ \cos(A-B) = \cos(225^\circ - 135^\circ) = \cos(90^\circ) = 0 \] (b) For \( A = 69^\circ \) and \( B = 30^\circ \): \[ \cos A \cos B = \cos(69^\circ) \cdot \cos(30^\circ) \approx 0.3584 \cdot 0.8660 \approx 0.309 \] For \( A = 225^\circ \) and \( B = 135^\circ \): \[ \cos A \cos B = \cos(225^\circ) \cdot \cos(135^\circ) = (-\frac{1}{\sqrt{2}})(-\frac{1}{\sqrt{2}}) = \frac{1}{2} = 0.5 \] (c) For \( A = 69^\circ \) and \( B = 30^\circ \): \[ \cos A \cdot \cos B + \sin A \cdot \sin B \approx 0.309 + 0.933 + 0.5 \approx 1.242 \text{ (not a valid range for cosine)} \] For \( A = 225^\circ \) and \( B = 135^\circ \): \[ \cos A \cdot \cos B + \sin A \cdot \sin B = \left(-\frac{1}{\sqrt{2}}\right) \left(-\frac{1}{\sqrt{2}}\right) + \left(-\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) = \frac{1}{2} - \frac{1}{2} = 0 \] And there you have it! Just plug in the values into your calculator for a quick and accurate analysis! Happy calculating!
