Move the slider to change the radius length.[br]Move point P until the measure of the radius is the same as the length of the arc.

1. What are the measures of the radius and the purple arc?[br]2. How many times larger is the arc of a 360 degree angle than the radius?[br]3. Move point P until the measure of the arc equals the radius. This measure correspondes to 1 radian.[br]4. Move point B and observe: What measurements change? What measurements stay the same?[br]5. What angle is bigger, one that measures 1 radian or one that measures 60 degrees?[br]6. Imagine an angle that measures 2 radians, how many degrees would that be? How about 5 radians? 1/2 radian? Check your answers in the construction.[br]7. In your own words, what is a radian?[br]8. Imagine a 180 degrees angle, what is the measure in radians? Is this reasonable? How many radians is a 90 degree angle? A 360 degree? why?

Have you ever wondered if a "tangent line" in Geometry has any relation to the "tangent of an angle" in trigonometry?[br][br]Or "secant line" and "secant of an angle?"[br][br]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?)[br][br]This worksheet is intended to help students see such connections.

[br]Drag the red point along the circumference of the unit circle, observing the changes in the various line segments.[br][br]Toggle on/off the various checkboxes and adjust the slider.[br][br]In Geometry, what does it mean for a line or line segment to be tangent to a circle?[br][br]In Geometry, what does it mean for a line or line segment to be secant to a circle?[br][br]Thinking of the Geometry definitions, why are the terms "tangent" and "secant" appropriate for labeling the respective line segments on the unit circle?[br][br]Referencing the unit circle, why is tan(θ) equal to the quotient of sin(θ)/cos(θ)?[br][br]In the unit circle what angle is formed by the line segments representing sine and cosine?[br][br]In the unit circle what angle is formed by the line segments representing tangent and cotangent?[br][br]There are several similar/proportional right triangles on the unit circle at any time (Except arguably the case when θ is a "quadrantal" angle). Using proportions, explain why any trig function(θ) = cofunction(complement of θ). In other words,[br][list][*]sine(θ) = cosine(complement of θ) for all values of θ[/*][*]tangent(θ) = cotangent(complement of θ) for all values of θ[br][/*][*]secant(θ) = cosecant(complement of θ) for all values of θ[/*][/list]