1.) Tangent?
Tangent is equal to sine/cosine, according to the Ratio Identity for tangent anyway. An asymptote is created wherever the denominator of a ratio equals 0 (and therefore is undefined), so whenever cosine's y value equals 0 on the graph, there will be an asymptote.
As you can see in this graph of cosine, the points where the y value is zero are -3pi/2, -pi/2, pi/2, and 3pi/2 (there are more, but let's not worry about that.)
In this graph of tangent, the points where the asymptotes are located are -3pi/2, -pi/2, pi/2, and 3pi/2 (again, there are more, but this is just a snapshot.)
2.) Cotangent?
Cotangent is equal to cosine/sine, also according to the Ratio Identity for Tangent. As stated before, an asymptote is found where the denominator equal 0. In this case, the denominator is sine, so the asymptotes are wherever sine's y value equals 0 on the graph.
As you can see in this graph of cosine, the points where the y value is zero are -2pi, -pi, 0, pi, and 2pi.
In this graph of cotangent, the points where the asymptotes are located are 0, 3.14, and 6.28- all of which are (approximately) the same points where sine equals 0.
3.) Secant?
Secant is equal to 1/cos, according to the Reciprocal Identity for secant. (We're done with Ratio Identities for now.) Secant has asymptotes too, which are located at the beginning and the end of each period.
Additionally, the minima and/or maxima of the secant graph also corresponds with the peaks and valleys of the cosine graph.
4.) Cosecant?
Cosecant is equal to 1/sin, according to the Reciprocal Identity for cosecant. Cosecant also has asymptotes (what a surprise), which are located at the beginning and the end of each period.
Additionally, the minima and/or maxima of the cosecant graph also corresponds with the peaks and valleys of the sine graph.
Resources:
http://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html
http://www.purplemath.com/modules/triggrph3.htm
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