# Question Video: Using the Unit Circle to Explain Symmetry of the Trigonometric Functions Mathematics • 10th Grade

Which symmetry of the unit circle proves that the tangent function is odd?

02:28

### Video Transcript

Which symmetry of the unit circle proves that the tangent function is odd?

Letβs start by reminding ourselves of the definition of an odd function. A function is odd if and only if π of negative π₯ is equal to negative π of π₯. 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 π. At its most basic level, it can tell us whether the values of sin, cos, and tan π in each quadrant are positive or negative.

Letβs take the tangent function. We can construct a right-angled triangle in the first quadrant using an ordered pair π and π, where both π and π are positive real numbers. The opposite side of this right-angled triangle, represented by π in our ordered pair, is positive, and the adjacent, represented by π, is also positive. Tan π is equal to π over π, and a positive divided by a positive gives us a positive solution. For this value of π, tan π is positive.

Remember, an odd function is one for which π of negative π₯ is equal to negative π of π₯. Weβve already worked out π of π₯, so letβs look at a negative value of π. For this ordered pair, which is a reflection in the π₯-axis of the previous one, π is now negative, as weβre measuring it in a clockwise direction. But what about tan π?

Well, the opposite side is now negative, since the π¦ value in our ordered pair is negative π. The adjacent however is still positive, since π₯ in our ordered pair remains positive π. In this case, tan π is equal to negative π divided by positive π, which is negative. By considering the ordered pair which we reflected in the π₯-axis, weβve proven that π of negative π₯ is equal to negative π of π₯. The tangent function is odd. Therefore, symmetry about the π₯-axis proves that the tangent function is odd.