BQ #3 : Unit T Concepts 1-3 - How do the graphs of sine and cosine relate to each of the others?

How Do the Graphs of Sine and Cosine Relate to Each of the Others?

TANGENT

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     The trig ratio as we all know is sine over cosine. We also know that on a Unit Circle, sine is positive on Quadrants l and ll, cosine is positive on l and lV, and tangent is positive on l and lll. On a graph, sine and cosine have a period of 2pi, while tangent only has 1pi. On the first quadrant (the red shaded section from the snapshot pic), both sine (the red graph) and cosine (the green graph) are positive. When plugged into the ratio to get tangent, algebraically it should be positive as well; and on the graph, too, it shows the tangent graph (the orange graph) positive. On the second quadrant (the green shaded section), sine is only positive while cosine is negative. As a ratio, tangent would be negative, which is proven on the graph as well. For quadrant three (the light orange section), both sine and cosine are negative but together make a positive tangent. As noticed on the graph, tangent is on the positive section in the third quadrant. And finally on the fourth quadrant (blue section), only cosine is positive and creates a negative tangent, which on the graph also shows.
     Tangent is different from sine and cosine as it has asymptotes at pi/2 and 3pi/2. Asymptotes are created when when a fraction, or ratio, has a denominator of 0. In this case, cosine has to be 0; and on a Unit Circle, sine is 0 at pi/2 (90 degrees) and 3pi/2 (270 degrees). On the graph, tangent will move closer to these special restrictions but never actually reach them.

COTANGENT

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     Cotangent is the inverse of tangent, meaning the ratio will be cosine over sine. With the previous information from before, both sine and cosine are positive in the first quadrant and will result with a positive cotangent, which is shown on the graph from the screenshot pic. On the second quadrant, cosine is negative while sine is positive, which will result with a negative cotangent on the graph (along with cosine). Moving to the third quadrant, both sine and cosine are negative which created a positive cotangent and will graphically be above the other two. Finally on the fourth quadrant, cosine is positive but with a negative sine, cotangent is negative and underneath with sine. 
     Cotangent as well contains aymptotes, however, at pi and 2pi. For cotangent and because of its inverse ratio, sine has to equal to 0 in order for the asymptotes to exists. On the Unit Circle, sine is 0, pi (180 degrees), and back to 0 which is also 2pi (360 degrees). On the graphs, the cotangent will never reach the asymptotes but it will get closer and closer to it. 

SECANT

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     Secant is the inverse of cosine, which also has a period of 2pi and start at the amplitudes. If we remember, cosine and secant are positive at l and lV and negative at ll and lll. When graphing them, it shows and profs the same results. On quadrant one and four, the secant graph (the purple graph) is above with the positive cosine but going in an inverse direction. For quadrants two and three, they are under the negative cosines and as well going on the inverse direction. 

     Secants is also a trig ratio with asymptotes and is seen through its ratio 1/cosine. Cosine (the denominator in this case) needs to equal 0 again which is found in pi/2 and 3pi/2, which on the unit circle it would have been 90 and 270 degrees. 

COSECANT

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     Cosecant is the inverse of sine, which has a period of 2pi. Using the Unit Circle, sine and cosecant are positive at l and ll while negative at lll and lV. Graphing cosecants would be somewhat similar to graphing sine. It would be positive on the first and fourth and have a parabola at the peak of the amplitudes with an inverse direction. It would be negative at lll and lV and underneath of the sine graph with same parabolas: negative and heading at an inverse direction of the amplitude. 

     Cosecants like secants, tangent, and cotangents, have asymptotes. The trig ratio is 1/sine at which sine has to equal 0. Sine is only negative at 0, pi (180 degrees), and 2pi (360 degrees and same as 0). At these asymptotes, cosecant graphs would not be able to overlap them but only get nearer.