Sketch the graphs of y = Sin x and y = pi/2 - 2x for 0 <= x <= pi/2 on the same axis

Draw an x-axis and a y-axis. Label the origin $(0,0)$. Mark $\frac{\pi}{2}$ on the x-axis. For the y-axis, mark 1, $\frac{\pi}{2}$ (approximately 1.57), and $-\frac{\pi}{2}$ (approximately $-1.57$).

• At $x=0$, $y = \sin(0) = 0$. Plot the point $(0,0)$.
• At $x=\frac{\pi}{2}$, $y = \sin(\frac{\pi}{2}) = 1$. Plot the point $(\frac{\pi}{2}, 1)$.
• Draw a smooth curve connecting these two points, starting from $(0,0)$ and increasing to $(\frac{\pi}{2}, 1)$. This is the first quarter of a sine wave.

This is a straight line.

• At $x=0$, $y = \frac{\pi}{2} - 2(0) = \frac{\pi}{2}$. Plot the point $(0, \frac{\pi}{2})$.
• At $x=\frac{\pi}{2}$, $y = \frac{\pi}{2} - 2(\frac{\pi}{2}) = \frac{\pi}{2} - \pi = -\frac{\pi}{2}$. Plot the point $(\frac{\pi}{2}, -\frac{\pi}{2})$.
• Draw a straight line connecting these two points: $(0, \frac{\pi}{2})$ and $(\frac{\pi}{2}, -\frac{\pi}{2})$.

The sketch will show the sine curve starting at the origin and rising to $( \frac{\pi}{2}, 1)$, and a straight line starting at $(0, \frac{\pi}{2})$ on the y-axis and descending to $(\frac{\pi}{2}, -\frac{\pi}{2})$ on the x-axis. The two graphs will intersect at some point within the interval $0 < x < \frac{\pi}{2}$.