Trigonometry
Trigonometry is often visualized with a circle of radius 1 (the unit circle), and most students are familar with the cosine and sine of an angle from the positive x-axis as the x and y coordinates of the point on the circle intersected by the other ray of the angle. The cosine and sine are often visualized as the legs of the right triangle defined by the origin, the point of intersection, and the point on the x-axis under the intersection point.
Less familiar are the two triangles similar to this one which help visualize the other basic trig functions: tangent, secant, cotangent and cosecant.
The visualization below looks intimidating at first but it's not hard to work out all the relationships by remembering the definitions of these other functions in terms of the cosine and sine and also keeping aware that:
- The triangles are similar so the sides are in proportion and
- One of the sides of each triangle has length 1.
Can you see:
- Where the names tangent, cotangent, secant and cosecant come from?
- Why the three Pythagorean trig identities are true?
Try it! Drag the green intersection point to change the angle and see what happens.
Jon Dreyer • Math tutor • Computer Science tutor
781-696-2614 •
81 Baker Ave,
Lexington MA 02421-6228 • email
