1. Sketch the curve \( y=\sin x \) for \( 0^{\circ} \leq x \leq 360^{\circ} \).

To sketch the curve \( y=\sin x \) for \( 0^{\circ} \leq x \leq 360^{\circ} \), start by plotting key points. At \( x = 0^{\circ} \), \( \sin 0^{\circ} = 0 \); at \( 90^{\circ} \), \( \sin 90^{\circ} = 1 \); at \( 180^{\circ} \), \( \sin 180^{\circ} = 0 \); at \( 270^{\circ} \), \( \sin 270^{\circ} = -1 \); and at \( 360^{\circ} \), \( \sin 360^{\circ} = 0 \). Now these points outline a smooth, wavy curve that rises to 1 at \( 90^{\circ} \), dips to -1 at \( 270^{\circ} \), and returns to 0 at \( 360^{\circ} \). The curve oscillates between -1 and 1, creating the familiar sine wave shape, which smoothly transitions through the key points. This curve appears frequently in mathematical models, especially in physics and engineering! Think of it as representing sound waves, alternating current in electricity, or even the motion of a pendulum. Its periodic nature makes it a fantastic tool for modeling anything that oscillates. So next time you hear a beautiful sound wave or feel an electric current, remember—it's all about \( y = \sin x \)!