Unit T Big Question #4:

4. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain. 

Sine and cosine do not have asymptotes because of their different unit circle ratios.
Sine = y/r & cosine = x/r. R will always equal 1.

Ratio chart:

This chart displays the ratios of all the trig functions.  Because 'r' will always equal 1, there is no way that sine and cosine can ever be undefined & have asymptotes. Instead,they have amplitudes. 

However, the other trig functions do have asymptotes b/c their denominator is capable of equaling to 0 -> causing the trig function to become undefined.

csc: r/y -> r/0 -> undefined -> asymptote

sec: r/x -> r/0 -> undefined -> asymptote

tan: y/x -> y/0 -> undefined -> asymptote 

cot: x/y -> x/0 -> undefined -> asymptote

Tangent's asymptotes would be where cosine = 0 (pi/2 & 3pi/2)

Cotangent's asymptotes would be where sine = 0 (0 & pi)

Sin(x) graph: 

Cos(x) graph: 

These images display the sine & cosine graphs and their amplitudes, not asymptotes.

Citations: Unit Circle ratio chart: www.kirchmathanalysis.blogspot.com
Sin/cos graphs: http://www.intmath.com/trigonometric-graphs/4-graphs-tangent-cotangent-secant-cosecant.php