BQ#3 – Unit T Concepts 1-3

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

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In the graph above, we see three different three different lines: sine(red), cos(blue), and tan (orange). In Quadrant I (the blue section), we see that both the sine and cosine values are positive. Looking back at our identities we know that, tangent equals sine/cosine. if sine and tangent are positive, they will continue to yield positive results. Therefore, tangent is positive in the first quadrant. In the second quadrant, sine is positive and cosine is negative. Knowing that tan = sin/cos, then tangent is going to have to be negative. In quadrant three both sine and cosine are negative. If we divide a negative by a negative we will come to find a positive answer. In quadrant IV, we have sin being negative and cos being positive. If we divide a negative by a positive, tangent is going to be negative. To find our asymptotes, we have to figure out where cosine will equal "0." By looking at the graph we see that cosine equals zero at (pi/2, 0) and (3pi/2, 0). We can see this asymptote as the orange line can not touch the points given. 

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In quadrant I, both sine and cosine are positive; this will lead to positive cotangent! Having a positive cotangent creates a graph that starts from the axis and goes up. From here, we should think of our identities. We know that "cot = cos/sin." This means that we will have asymptotes where sin = 0 (0 on bottom = undefined = asymptote). In Quad II, sin is positive while cos is negative. Therefore, if we divide cos by sin, we will receive a negative cotangent. This leads to a negative shift in the graph. In Quad III, cotangent is going to be positive because sin and cos are both negative. As we all know, two negatives make a positive. The final quadrant involves a negative sin value and a positive cos value. Dividing the two together would result in a negative turn in the graph. 

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In Quadrant I cosine and secant are both positive. Thanks to our identities, we know that secant is the reciprocal of cosine. So where cos is (0,1), secant is going to be the reciprocal of the value (1). Since we know that secant is the  reciprocal of cos, we know that the asymptotes will be located where cos = 0. In Quadrant II, cosine is negative so secant is negative. We must not forget that the ACTUAL graph will not make contact with the asymptote boundaries. We can see that cosine is positive in Quadrant IV. Therefore, if we follow our ongoing trend, secant is going to have to be positive. We draw the graph from the high and low points of the valleys. These types will always be parabolas that have their set boundaries.

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For our cosecant graph, asymptotes and sine values are VERY important. Wherever sin(x) = 0 there will be asymptotes for the cosecant graph. In quadrant 1, our sine value is positive; the same goes for quadrant II. So if sin is positive in both quadrants that means that cosecant will be above the x-axis (we draw csc from the vertex). In quadrant III and IV, we see that our sine value is negative. This means that our cosecant graph will also be negative. As you can see, the entire cosecant graph depends on the vertex and asymptotes of the sine graph. Why does this happen? We know that cosecant = 1/sin, wherever sin(x) = 0. This occurs because when sin=0 we receive the asymptotes that set our cosecant graph's boundaries.



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