BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each other the others? Emphasize asymptotes in your response.

a. Tangent? The graph of tangent has its asymptotes based on cosine. To understand how this works, one must know that asymptotes are found in a graph when the trigonometric function is undefined. The trig ratio for tangent is sine/cosine, and we know that it is undefined when the denominator is equal to zero. So, if cos(x)=0, tangent becomes undefined, and thus it has asymptotes. Cosine equals zero at pi/2 and 3pi/2, so we know that two of our asymptotes lie there. Also, the positive and negative values of tangent are influenced by sine and cosine. For example, in quadrant two, sine is positive while cosine is negative. When we divide sine by cosine, since it is tangent's ratio, we get a negative value. Thus, we find that tangent is negative in quadrant two, which is true. This method applies for each quadrant around the unit circle.

b. Cotangent? The graph of cotangent has its asymptotes based on sine. The trig ratio for cotangent is cosine/sine, and we know that it is undefined when the denominator is equal to zero (thus giving us asymptotes). So, if sine(x)=0, cotangent becomes undefined, and then it has asymptotes. Sine equals zero at 0 and pi, so we know that two of our asymptotes lie there. Also, the positive and negative values of cotangent are influenced by sine and cosine because of its ratio. For example, in quadrant three, both sine and cosine are positive. When we divide cosine by sine, since it is cotangent's ratio, we get a positive value (a negative divided by a negative equals a positive). Thus, we find that cotangent is positive in quadrant three, which turns out to be true. This method applies for each quadrant around the unit circle.

c. Secant? The graph of secant has its asymptotes based on cosine, since it is its reciprocal. The trig ratio for secant is 1/cosine, and we know that it becomes undefined when the denominator is equal to zero (thus giving us asymptotes). So, if cosine(x)=0, secant becomes undefined, and then it has its asymptotes. Cosine equals zero at pi/2 and 3pi/2, so we know that two of our asymptotes lie there. Also, the positive and negative values of secant are also influenced by cosine, since they share the same ones.

d. Cosecant? The graph of cosecant has its asymptotes based on sine, since it is its reciprocal. The trig ratio for cosecant is 1/sine, and we know that it becomes undefined when the denominator is equal to zero (thus giving us asymptotes). So, if sine(x)=0, cosecant becomes undefined, and there we have our asymptotes. Sine equals zero at 0 and pi, so we know that we have two of our asymptotes there. Also, the positive and negative values of secant are also influenced by sine, since they share the same ones.