- How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.
- For these graphs we will be looking at the quadrants and the relations they have to our sine and cosine graphs. We will also be using a little bit of identities and asymptote knowledge in order to find the special relation.
Sine and cosine are related to the tangent graph when we are looking at the sign of the tangent graph. We know that tangent equals sine/cos as a ratio identity so if one of the graphs is negative then tangent will be negative. If both of the graphs are either positive or negative then the tangent graph will be positive.
If we look at this picture, the pattern fits perfectly for each quadrant where there is sometimes a switch in positive or negative. Watch as the tangent graph goes up or down according to its asymptotes as well.
This is the same pattern for cotangent where if one is negative than cotangent is also negative. The asymptotes are in a different location but this will be gone into further detail in BQ#4.
We are going to be looking at something different for secant and cosecant graphs. For these graphs we will only look at one of the sine and cosine graphs to find the relation between them. For a secant graph, it is related to a cosine graph because the reciprocal of cosine is secant. We can also use this information to spot where the asymptotes will be so in this case the asymptotes will be where cosine equals zero.
What we can also see from this picture is that as the cosine value on the graph goes towards zero, the secant graph will spike up to ridiculous numbers. This is because the reciprocal of a very small number or a fraction in this case equates to a very large number. We also see that when the y value equals 1 or -1 on the cosine graph, the secant graph also has that same value because the reciprocal of 1 is 1.
Now for a cosecant graph, the partner we should be focusing on is the sine graph and the same reciprocal rule still applies. Of course the asymptotes will now be in different positions because we are dealing with a different graph but the same rules still apply as seen in this picture.
- All images screenshot from https://www.desmos.com/calculator/hjts26gwst