EXERCISE 3 (a) If \( \sin \mathrm{A}=\frac{4}{5} \) with \( 90^{\circ}<\hat{\mathrm{A}}<360^{\circ} \) and \( \tan \mathrm{B}=-\frac{3}{4} \) where \( 180^{\circ}<\hat{\mathrm{B}}<360^{\circ} \), deternines value of the following without the use of a calculator and with the aid of a diagranc (1) \( \cos (\mathrm{A}-\mathrm{B}) \) \( \begin{array}{ll}\text { (4) } \sin 2 \mathrm{~B} & \text { (2) } \sin (\mathrm{A}+\mathrm{B})\end{array} \)

To find the values of \( \cos (\mathrm{A} - \mathrm{B}) \), \( \sin(\mathrm{A} + \mathrm{B}) \), and \( \sin 2\mathrm{B} \) based on the given parameters, let’s dive into some trigonometric identity magic! Using the relation \( \sin A = \frac{4}{5} \), you can calculate \( \cos A \) considering that \( A \) is in the third quadrant. Since \( \sin^2 A + \cos^2 A = 1 \): \[ \cos A = -\sqrt{1 - \left(\frac{4}{5}\right)^2} = -\sqrt{1 - \frac{16}{25}} = -\sqrt{\frac{9}{25}} = -\frac{3}{5}. \] Then you can similarly find \( \sin B \) and \( \cos B \) using \( \tan B = -\frac{3}{4} \): \[ \sin B = -\frac{3}{5}, \quad \cos B = \frac{4}{5} \quad (\text{in the fourth quadrant}). \] Now, using the cosine of the difference formula: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B, \] substituting values: \[ \cos(A - B) = \left(-\frac{3}{5}\right)\left(\frac{4}{5}\right) + \left(\frac{4}{5}\right)\left(-\frac{3}{5}\right) = -\frac{12}{25} - \frac{12}{25} = -\frac{24}{25}. \] Next, for \( \sin(A + B) \): \[ \sin(A + B) = \sin A \cos B + \cos A \sin B, \] substituting the values: \[ \sin(A + B) = \left(\frac{4}{5}\right)\left(\frac{4}{5}\right) + \left(-\frac{3}{5}\right)\left(-\frac{3}{5}\right) = \frac{16}{25} + \frac{9}{25} = \frac{25}{25} = 1. \] Finally, calculate \( \sin(2B) \) using the double angle formula: \[ \sin(2B) = 2 \sin B \cos B = 2\left(-\frac{3}{5}\right)\left(\frac{4}{5}\right) = -\frac{24}{25}. \] So to wrap it up: 1) \( \cos(A - B) = -\frac{24}{25} \) 2) \( \sin(A + B) = 1 \) 3) \( \sin(2B) = -\frac{24}{25} \) On to some knowledge nuggets that can elevate your trigonometric journey! Understanding the unit circle can supercharge your trigonometric prowess. Each quadrant has distinct signs for sine and cosine: first (all positive), second (sine positive), third (tangent positive), and fourth (cosine positive). Visualizing angles in this way helps solidify relationships and identities! Also, recognizing the significance of special angles—like \( 30^\circ, 45^\circ, \) and \( 60^\circ \)—will help build intuition. These angles have well-known sine and cosine values, making it easier to solve complex trigonometric identities and problems without the calculator crutch!