Given: \( \underline{\sin \alpha=\frac{3}{5} \text { and } 90^{\circ}<\alpha<270^{\circ}} \) With the aid of a sketch and without using a calculator, determine: (a) \( \tan \alpha \) (b) \( \cos \left(90^{\circ}+\alpha\right) \)

Let's solve each part step by step using trigonometric identities and the given information. ### Given: - \( \sin \alpha = \frac{3}{5} \) - \( 90^{\circ} < \alpha < 270^{\circ} \) (This places \( \alpha \) in the **second** or **third** quadrant.) ### (a) Determining \( \tan \alpha \) 1. **Find \( \cos \alpha \):** Using the Pythagorean identity: \[ \sin^2 \alpha + \cos^2 \alpha = 1 \] \[ \left(\frac{3}{5}\right)^2 + \cos^2 \alpha = 1 \Rightarrow \frac{9}{25} + \cos^2 \alpha = 1 \] \[ \cos^2 \alpha = 1 - \frac{9}{25} = \frac{16}{25} \] \[ \cos \alpha = \pm \frac{4}{5} \] **Determine the sign of \( \cos \alpha \):** - Since \( \alpha \) is in the **second** quadrant (\( 90^{\circ} < \alpha < 180^{\circ} \)), where cosine is negative: \[ \cos \alpha = -\frac{4}{5} \] 2. **Calculate \( \tan \alpha \):** \[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4} \] ### **Answer for (a):** \[ \tan \alpha = -\frac{3}{4} \] --- ### (b) Determining \( \cos \left(90^{\circ} + \alpha\right) \) Use the cosine of a sum identity, particularly for angles involving \( 90^{\circ} \): \[ \cos(90^{\circ} + \alpha) = -\sin \alpha \] Given \( \sin \alpha = \frac{3}{5} \): \[ \cos(90^{\circ} + \alpha) = -\frac{3}{5} \] ### **Answer for (b):** \[ \cos \left(90^{\circ} + \alpha\right) = -\frac{3}{5} \] Answer: (a) \( \tan \alpha = -\dfrac{3}{4} \); (b) \( \cos(90^\circ + \alpha) = -\,\dfrac{3}{5} \)