All Six Trig Functions

Pre-Calculus · Axiom Academy

Sine and cosine are the two coordinates of a point on the unit circle — the other four functions are just ratios and flips of those two. 1. Tangent Is the Slope of the Radius Drop a point at angle on the unit circle. Its coordinates ARE two of our functions: the x -coordinate is and the y -coordinate is . The third primary function, tangent , is their ratio — which is exactly the slope of the radius drawn out to that point: rise over run, y over x . x = cos θ is the run · y = sin θ is the rise tangent = rise ÷ run = the slope of the radius 2. The Reciprocal Functions Flip the Fraction Every primary function has a partner you get by flipping its fraction over: 1 divided by it. These are cosecant , secant , and cotangent . The animation pairs each bar with its reciprocal — watch a tall bar flip to a short partner and a short bar flip to a tall one, because always. , and the same for the other two pairs — that is what "reciprocal" means. Co sine and co secant are NOT reciprocals of each other. Cosecant is the reciprocal of sine ; secant is the reciprocal of cosine . The matching prefixes are a coincidence of naming, not a pairing.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.