BQ#3 – Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?

a. Tangent

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In the last unit, we learned about ratio identities. The ratio identity of tanθ = sinθ/cosθ. The asymptotes are where the graph is undefined, so cos would have to equal 0 for this to be true. Also, cos=0 at 90 degrees and 270 degrees, which is pos/neg pi/2. This is where the asymptotes are located. Moreover, when sine crosses the x axis, tangent does not because there is an asymptote. When cosine crosses the x-axis, there is another asymptote for tangent. A tangent graph is not continuous because there are asymptotes restricting it.

b. Cotangent

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The ratio identity of cotθ = cosθ/sinθ. Therefore,in order for there to be asymptotes, sinθ must equal 0. Therefore, there are asymptotes at 0 degrees and 180 degrees, which is 0 and pi. Thus, similarly to tangent, the asymptotes for cotangent correspond to where sine and cosine cross the x axis. It is also not continuous.

c. Secant

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In a secant graph, they form parabolas above the mountains and below the valleys of a cosine graph. Moreover, it is not continuous because there are asymptotes.

d. Cosecant

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.A cosecant graph corresponds with a sine graph. A cosecant graph also forms parabolas like a secant graph, but this time they touch the mountains and valleys of a sine graph. This graph is also not continuous because of the asymptotes.