Which symmetry of the unit circle proves that the tangent function is odd?
Let’s start by reminding ourselves of the definition of an odd function. A function is odd if and only if 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. So how do we use the unit circle to prove that the tangent function is odd?
Remember, the unit circle is a circle with a radius of one that allows us to evaluate sin, cos, and tan for specific values of 𝜃. At its most basic level, it can tell us whether the values of sin, cos, and tan 𝜃 in each quadrant are positive or negative.
Let’s take the tangent function. We can construct a right-angled triangle in the first quadrant using an ordered pair 𝑎 and 𝑏, where both 𝑎 and 𝑏 are positive real numbers. The opposite side of this right-angled triangle, represented by 𝑏 in our ordered pair, is positive, and the adjacent, represented by 𝑎, is also positive. Tan 𝜃 is equal to 𝑏 over 𝑎, and a positive divided by a positive gives us a positive solution. For this value of 𝜃, tan 𝜃 is positive.
Remember, an odd function is one for which 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. We’ve already worked out 𝑓 of 𝑥, so let’s look at a negative value of 𝜃. For this ordered pair, which is a reflection in the 𝑥-axis of the previous one, 𝜃 is now negative, as we’re measuring it in a clockwise direction. But what about tan 𝜃?
Well, the opposite side is now negative, since the 𝑦 value in our ordered pair is negative 𝑏. The adjacent however is still positive, since 𝑥 in our ordered pair remains positive 𝑎. In this case, tan 𝜃 is equal to negative 𝑏 divided by positive 𝑎, which is negative. By considering the ordered pair which we reflected in the 𝑥-axis, we’ve proven that 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. The tangent function is odd. Therefore, symmetry about the 𝑥-axis proves that the tangent function is odd.