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The Tangent Function

The tangent function, , can be derived as the ratio of to . There is another way to look at it, and it helps explain the name "tangent" at the same time. Construct a vertical line tangent to the Unit Circle at the point . Now extend the radius at any angle until it intersects this line. The length of the segment between and this intersection is the value of . When or , the radius is parallel to the tangent line, and thus they never intersect. The segment is therefore actually a ray with infinite length; thus at these angles.