The tangent function, [math]tan\left(\theta\right)[/math], can be derived as the ratio of [math]sin\left(\theta\right)[/math] to [math]cos\left(\theta\right)[/math]. There is another way to look at it, and it helps explain the name "tangent" at the same time.[br][br]Construct a vertical line [i]tangent to[/i] the Unit Circle at the point [math]\left(0,1\right)[/math]. Now extend the radius at any angle [math]\theta[/math] until it intersects this line. The length of the segment between [math]\left(0,1\right)[/math] and this intersection is the value of [math]tan\left(\theta\right)[/math].[br][br]When [math]\theta=\frac{\pi}{2}[/math] or [math]\theta=\frac{3\pi}{2}[/math], the radius is parallel to the tangent line, and thus they never intersect. The segment is therefore actually a ray with infinite length; thus [math]tan\left(\theta\right)\longrightarrow\infty[/math] at these angles.