BQ #4: Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill? Use unit circle ratios to explain.

If we refer to the unit circle and ASTC, we know that tangent and cotangent are positive in quadrants 1 and 3 and negative in quadrants 2 and 4. If we were to break apart the unit circle into a line, we would get + - + - values for both trig functions. And unlike the other trig functions, tangent and cotangent have a period of pi rather than 2pi.

Tangent has the ratio of sin/cos (y/x), so we get asymptotes whenever cosine is equal to zero. On the unit circle, that would be the values of pi/2 and 3pi/2. If we were to draw out the graph, we notice that it goes "uphill" because each asymptote compresses an area where the graph would start with negative values and into positive values.

Cotangent has the ratio of cos/sin (x/y), so we get asymptotes whenever sine is equal to zero. On the unit circle, that would be the values of 0 and pi. If we were to draw out the graph, we would notice that it goes "downhill" because each asymptote compresses an area where the graph would start with positive values and into negative values.