Symmetry and Concavity of Sinusoids

Sine and cosine come from the same unwrapped circle, but they land on the graph differently because of one algebraic fact: sin(-theta) = -sin(theta), while cos(-theta) = cos(theta). Sine is odd, so its graph has 180-degree rotational symmetry about the origin — spin the curve halfway around and it lands back on itself. Cosine is even, so its graph is a mirror image across the y-axis instead. You can see this directly from the unit circle: reflecting an angle theta to -theta flips the y-coordinate (sine) but leaves the x-coordinate (cosine) alone.

This symmetry connects directly to concavity. Trace y = sin(x) from x = 0 to x = 2pi. Between 0 and π, the curve bulges downward in the first half and upward in the second — concretely, it is concave down from 0 to π and concave up from π to 2pi. The switch happens exactly at x = 0, π, and 2pi, which are precisely the points where the graph crosses its own midline, y = 0.

That pattern is not a coincidence. Every time a sinusoid crosses its midline, concavity changes sign, and a sign change in concavity is the definition of a point of inflection. So for any sine or cosine curve, the midline crossings are always the inflection points, and the max and min points (where the curve is farthest from the midline) are always where concavity is most extreme in one direction. This links symmetry, concavity, and the key points of a sinusoid into one consistent picture, which matters once you start stretching and shifting these graphs.