2. How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.
a. Tangent? Tan = sin/cos. When cosine is zero, it makes tangent become undefined -> thus, it will have an asymptote on the points where cos = 0: pi/2 or 3pi/2.
b. Cotangent? Cotangent is the exact reciprocal of tangent. Cotangent = cos/sine. When sine is zero, it causes cotangent to become undefined. Thus, it will have an asymptote when sine is 0 or pi.
c. Secant? Secant is the reciprocal of cosine -> 1/cos. Thus, when cos = 0 -> it will be undefined & = asymptotes!
d. Cosecant? Cosecant is the reciprocal of sine -> 1/sin. Thus, when sine = 0 -> it will be undefined & = asymptotes!
Sine & cosine do not have asymptotes because of their ratios. Sine = y/r. Cosine = x/r. R will always equal 1 and thus it can never be undefined.
Tan(x) graph w/ asymptotes.
Cot(x) graph w/ asymptotes.
Csc(x) graph w/ asymptotes.
Sec(x) graph w/ asymptotes:
Sin(x) graph:
Cos(x) graph:
The images of these graphs display the placement of the asymptotes and the difference between cos/sin graphs & csc/sec/tan/cot graphs.
Citations:
Images of the cos/sin/csc/sec/tan/cot graphs: http://www.intmath.com/trigonometric-graphs/4-graphs-tangent-cotangent-secant-cosecant.php
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