Why is the tan function called tangent?
In this video, we explore why the tangent function is called "tangent," derived from the Latin word tangere, for "to touch." We create a diagram on the unit circle in the xy-plane. We pick a random point rotated counterclockwise by theta radians. The x and y coordinates of this point are (cos(theta), sin(theta)).
Next, we draw a tangent line at this point, which is perpendicular to the radius. Using the point-slope form, we derive the equation of the tangent line and find its x and y intercepts, providing geometric meaning to the secant and cosecant functions (sec and csc). We also break the tangent line into two segments, finding that their lengths correspond to the tangent and cotangent (tan and cot) functions using the Pythagorean theorem.
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