How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.
1. Tangent
In the first quadrant (shaded red) you can tell that the cosine values in green and the sine values in red, are all positive because they are above the x-axis. Since all trig values are positive in the first quadrant we can infer that tan (the orange line) will be and is positive within this quadrant. In the second quadrant cosine is negative (green line) and sine is positive (red line) and with the tan ratio being: tan(x)=sin(x)/cos(x), it leaves us with a positive divided by a negative, which we get a product of a negative value. That being why in this quadrant the tangent values fall below the x-axis. When looking at our third quadrant we can see that our cos and sine values are both below the x-axis meaning their negative, once again making tangent positive because negative divided by negative gives you a positive. The same goes on in the fourth quadrant as cos is positive and sine is negative giving us a tan of negative. Now when we look at the asymptotes involved with such trig functions by looking at our ratio identity for tan= sin/cos and see that cosine must equal 0 in order to get an undefined value. We get cosine to equal zero by looking at the unit circle pointing out those x-values in which there is a zero, those coordinates being (0,1) which is pi and (0, -1) in other words 3pi/2. Those spots being where our tangent graph seems as if to stop, that is because they are our asymptotes. As noticed a difference being that tan involves assymptotes unlike sin and cos.
2. Cotangent
With Cotangent having a ratio of cos/sin, the values of cos and sin affect where cotangent will lye whether below or above the x-axis. This was the same case when dealing with tangent, it's only going to be the opposite. As the asymptotes are located at 0 and 180 because y=0, as you can see we look at the y value this time not x. By using the trig ratio for cot (cos/sin) if both Sin and cos where to be positive in the first quadrant ( red ), we get a positive value for cot. When in quadrant 2 (green) we get a negative value for cot, because we positive/ negative. For the third quadrant (orange) we get a positive value for cot, positive/positive. In the fourth quadrant (blue),we get a negative value for cot because it's a Positive/negative. The graph for cotangent holding a pattern that is positive, negative, positive, negative as seen in the image above. The only times we see it break up is at the asymptotes that lye on 0 and 180.
In the first quadrant (shaded red) you can tell that the cosine values in green and the sine values in red, are all positive because they are above the x-axis. Since all trig values are positive in the first quadrant we can infer that tan (the orange line) will be and is positive within this quadrant. In the second quadrant cosine is negative (green line) and sine is positive (red line) and with the tan ratio being: tan(x)=sin(x)/cos(x), it leaves us with a positive divided by a negative, which we get a product of a negative value. That being why in this quadrant the tangent values fall below the x-axis. When looking at our third quadrant we can see that our cos and sine values are both below the x-axis meaning their negative, once again making tangent positive because negative divided by negative gives you a positive. The same goes on in the fourth quadrant as cos is positive and sine is negative giving us a tan of negative. Now when we look at the asymptotes involved with such trig functions by looking at our ratio identity for tan= sin/cos and see that cosine must equal 0 in order to get an undefined value. We get cosine to equal zero by looking at the unit circle pointing out those x-values in which there is a zero, those coordinates being (0,1) which is pi and (0, -1) in other words 3pi/2. Those spots being where our tangent graph seems as if to stop, that is because they are our asymptotes. As noticed a difference being that tan involves assymptotes unlike sin and cos.
(Mrs. Kirch creation on desmos)
With Cotangent having a ratio of cos/sin, the values of cos and sin affect where cotangent will lye whether below or above the x-axis. This was the same case when dealing with tangent, it's only going to be the opposite. As the asymptotes are located at 0 and 180 because y=0, as you can see we look at the y value this time not x. By using the trig ratio for cot (cos/sin) if both Sin and cos where to be positive in the first quadrant ( red ), we get a positive value for cot. When in quadrant 2 (green) we get a negative value for cot, because we positive/ negative. For the third quadrant (orange) we get a positive value for cot, positive/positive. In the fourth quadrant (blue),we get a negative value for cot because it's a Positive/negative. The graph for cotangent holding a pattern that is positive, negative, positive, negative as seen in the image above. The only times we see it break up is at the asymptotes that lye on 0 and 180.
3. Secant
When looking at secant and the relationship it holds among cosine and sine is that it's the reciprocal of cosine, having a ratio of 1/cos. With cosine being the denominator, when it does equal a zero making the ratio undefined we get a assymptotes. When looking at the unit circle cosine equals zero at (90*) pi/2 and 3pi/2 (270*). By looking at the graph it's seen how when cosine is positive so shall the secant be positive, and when negative it will be negative.
When looking at secant and the relationship it holds among cosine and sine is that it's the reciprocal of cosine, having a ratio of 1/cos. With cosine being the denominator, when it does equal a zero making the ratio undefined we get a assymptotes. When looking at the unit circle cosine equals zero at (90*) pi/2 and 3pi/2 (270*). By looking at the graph it's seen how when cosine is positive so shall the secant be positive, and when negative it will be negative.
4. Cosecant
The relationship that Cosecant has with sine and cosine is that it is the reciprocal of sine, having a ratio of 1 over sine. With sine being the denominator when it is zero it makes the ratio undefined giving us an asymptote. When taking a look at the unit circle, the ones with a zero on the x is zero and pi, meaning that Cosecant has asymptotes at 0 and pi. As seen in the graph when sine is positive so will the cosecant graph. Same goes when sine is negative, sine controlling whether the cosecant graph goes up hill or down hill.
- (Mrs. Kirch creation on desmos)
* as noticed in all of the graphs dealing with asymptotes the line never crosses or touches the asymptote. Always by it never through it. As seen in the image above csc line in red just goes beside it.
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