One of the nicest definitions of the six basic trig functions involves drawing an ray from the origin of a unit circle with its center at the origin.¹ Where the

One of the nicest definitions of the six basic trig functions involves drawing an ray from the origin of a unit circle with its center at the origin.¹ Where the ray intersects the circle at point A draw a vertical line perpendicular to the x axis. The height of the line segment from A to the x axis will be the sine of the angle between the ray and the x axis. The length of the line segment from the origin to where your vertical line hits the x axis will be the cosine. Now, draw a tangent line perpendicular to the x axis and find the point B where your ray intersects the tangent line. The length of the segment from the origin to B will be the secant, the length of the segment from B to the x axis will be the tangent.² Finally, draw the tangent line parallel to the x axis and find the intersection of the ray with that line at C. The length of the segment from C to the origin will be the cosecant and the segment from the y axis to C will be the cotangent. You can use your basic trig identities and knowledge of similar triangles to verify the relationships between the functions and the triangles. Angles outside the first quadrant will give signed values that make sense if you consider segments going down or left to be negative (but down and left is positive). ⸻ 1. I’m making reference to cartesian coordinates strictly for the sake of convenience since I’m using only words to describe a diagram. 2. I’m doing this from memory and really hoping I’m not mixing up the tangent, cotangent, secant and cosecant