-Tangent
-Cotangent
-Secant
-Cosecant
1) Tangent
Okay so the ratio identity for tangent we know is sine/cosine. Also, we are aware that we will only have asymptotes where we get an undefined answer meaning that we divide by zero and therefore, cosine would have to be zero in order to make this asymptotes and an undefined answer. Now, we will have asymptotes at the 90 and 270 degree, so at pi/2 and 3pi/2. This being said, we know that tangent cannot touch them no matter what. Now, there are way more asymptotes, they practically go on forever, yet we can just stick to some.
2) Cotangent
We can basically use the same knowledge from above with Tangent, yet now we know that cotangent is equaled to cosine over sine and so the asymptotes will be found where cosine is zero so at 90 and 270 degrees, or pi/2 and 3pi/2. And we remember that the graph cannot touch the asymptotes!
3) Secant
We know that secant is the reciprocal of cosine and the ratio is 1/cos. Secant as well has asymptotes that are found where cosine is equaled to zero as well.
4) Cosecant
Now, the same rules apply like with the reciprocals and stuff yet the asymptotes will be located in different places due to the different kind of graph. Cosecant is the reciprocal of sine so its ratio is 1/sin.
References:
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