Unit Circle - Trig functions vs Geometry definitions
Have you ever wondered if a "tangent line" in Geometry has any relation to the "tangent of an angle" in trigonometry? Or "secant line" and "secant of an angle?" What relation is implied by the prefix "co-" when discussing sine and cosine, tangent and cotangent, or secant and cosecant? (Did you know it has to do with "complementary" angles?) This worksheet is intended to help students see such connections.
Drag the red point along the circumference of the unit circle, observing the changes in the various line segments. Toggle on/off the various checkboxes and adjust the slider. In Geometry, what does it mean for a line or line segment to be tangent to a circle? In Geometry, what does it mean for a line or line segment to be secant to a circle? Thinking of the Geometry definitions, why are the terms "tangent" and "secant" appropriate for labeling the respective line segments on the unit circle? Referencing the unit circle, why is tan(theta) equal to the quotient of sin(theta)/cos(theta)? In the unit circle what angle is formed by the line segments representing sine and cosine? In the unit circle what angle is formed by the line segments representing tangent and cotangent? There are several similar/proportional right triangles on the unit circle at any time (Except arguably the case when theta is a "quadrantal" angle). Using proportions, explain why any trig function(theta) = cofunction(complement of theta). In other words, * sine(theta) = cosine(complement of theta) for all values of theta * tangent(theta) = cotangent(complement of theta) for all values of theta * secant(theta) = cosecant(complement of theta) for all values of theta