Trigonometric Ratios in a Circle

This demonstration shows the relationship of the six basic trigonometric ratios to triangles within a unit circle. A unit circle is a circle with a radius of one. Explore what happens when you select each of the three options. In each case, one of the sides of the triangle is the same length as the radius, and the other two sides are marked with a trigonometric ratio. Explore what happens when you change the size of the angle. Think about why the secant and cosecant can never be less than one.

Additional questions to think about: 1. What does "co-" generally mean? Why do you think that the cosine, cotangent, and cosecant have those names? 2. What is the tangent of a circle? If you're not sure, consult a mathematics reference source. How are the tangent of a circle and the tangent of a triangle related? 3. What is the secant of a circle? Again, if you're not sure, check a reference work. Based on what you know or have learned about the secant of a circle, will any circle secant work for this demonstration? Why or why not? 4. Usually, the definitions of these ratios include two sides of the triangle. For instance, sine is defined as "the length of the side opposite an angle to the hypotenuse." Why isn't that needed for a unit circle?