Select the correct option. What is the exact value of \( \sin 210^{\circ}+\sin 270^{\circ} \) ? \( -\frac{3}{2} \)

To find the value of \( \sin 210^{\circ} + \sin 270^{\circ} \), we first determine the sine values. The angle \( 210^{\circ} \) is in the third quadrant, where the sine is negative. The reference angle is \( 30^{\circ} \), so \( \sin 210^{\circ} = -\frac{1}{2} \). For \( 270^{\circ} \), which is directly down the y-axis, \( \sin 270^{\circ} = -1 \). Now, adding these together: \[ \sin 210^{\circ} + \sin 270^{\circ} = -\frac{1}{2} + (-1) = -\frac{1}{2} - 1 = -\frac{3}{2}. \] Thus, the exact value of \( \sin 210^{\circ} + \sin 270^{\circ} \) is indeed \( -\frac{3}{2} \). For a deeper dive into the sine function, remember that the sine wave oscillates between -1 and 1. In the unit circle, each angle corresponds to a specific point defined by (cosine, sine). Understanding this can help you visualize how these values fit together as you explore different angles and their sine values! You can also experiment with the angles using your unit circle knowledge! Plot angles in different quadrants to see how sine values change. A quick sketch of the sine function can make it clearer: you’ll see a wave-like pattern, helping you to feel how the values behave at crucial angles like 0°, 90°, 180°, and 270°!