How do graphs of sine and cosine relate to each of the others ? Emphasize asymptotes in your response.
Tangent?
Cotangent?
Secant?
Cosecant?
Cosecant
Cosecant is similar to sine.The ratio for cosecant is 1/sin and since sin is y/r cosecant is r/y.In the unit circle wherever sine is positive so is cosecant. So basically, since sin is positive so is cosecant in the first quadrant. Sine does not have any asymptotes because they only occur when there is an undefined answer. However, cosecant does have asymptotes at o and pi.
Secant
Secant is related to cosine. Acording to the unit circle cosine is poitive in the first last quadrants and negative in the second and third. Secant follows this as well. The asymptotes or where an undefined answer occurs are at pi over 2 and 3pi over 2. This is where cosine is 0, making the ratio undefined. Because secant is 1/cosine.
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Cotangent
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The identity for cotangent is cosine over sine or x/y. It is very similar to tangent. Cosine and sine are positive in the first quadrant because cosine/sine will be positive In the second quadrant, the sine is positive and cosine is negative, so cotangent is negative because a negative cosine divided by a positive(sign wise) will be negative. For the third quadrant, cosine and sine are negative, making cotangent positive because a negative divided by a neagtive will cancel out the signs and make the answer positive. In the last quadrant cosine isnpositive and sine is negative, making cotangent negative because, again, a neagtive and positive leaves a negative. Since asymptotes occur when 0 is below the divid]sion sine, sine must eqaul zero so that an undefined answer may occur and asymptotes are found. Therefore, we know that sine is equal to 0at 0 and pi.
Tangent
The identity for tangent is sine/cosine or y/x. If sine or cosine are negative, then tangent will be negative because a negative and positive sign make a negative answer. If sine or cosine are positive or both negative, tangent will be positive because a negative and negative sign will be positive, as they cancel out. In the first quadrant, sine and cosine are positive, so tangent is positive. In the second quadrant, cosine is negative, but sine is positive so it is negative.In the third quadrant, both sine and cosine are negative, making tangent positive as they both cancel out each others signs. The fourth quadrant is negative because cosine is positive, but sine is negative. Tangent has asymptotes when cosine is equal to 0 because cosine in the denominator. They are at pi/2 and 3pi/2
The identity for tangent is sine/cosine or y/x. If sine or cosine are negative, then tangent will be negative because a negative and positive sign make a negative answer. If sine or cosine are positive or both negative, tangent will be positive because a negative and negative sign will be positive, as they cancel out. In the first quadrant, sine and cosine are positive, so tangent is positive. In the second quadrant, cosine is negative, but sine is positive so it is negative.In the third quadrant, both sine and cosine are negative, making tangent positive as they both cancel out each others signs. The fourth quadrant is negative because cosine is positive, but sine is negative. Tangent has asymptotes when cosine is equal to 0 because cosine in the denominator. They are at pi/2 and 3pi/2
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