1. Tangent
With tangent, we know that the unit circle ratio would be sine over cosine. If we would put that into variables, y over x. When we look at how sine and cosine graphs relate to tangent, we notice that if both sine and cosine are positive, then tangent would be positive; if one is negative, then tangent is negative; if both are negative, then tangent is positive. That is basic division rules. Unlike sine and cosine, tangent graphs have asymptotes (which are areas that we cannot touch; undefined). What we need to realize is that the sine over cosine ratio is responsible for those asymptotes: if we divide any number by 0, it would come out as undefined. If cosine (x) were to be equal to 0, that would result in an asymptote. If we were to look at a unit circle, we should know that the x value would equal 0 at 90 degrees and 270 degrees. Converting back to the tangent graph and radians, we know that our asymptotes would be at pi/2 and 3pi/2.
2. Cotangent
With cotangent, we face the reciprocal unit circle ratio of cosine over sine, or x over y. When we look at how cosine and sine graphs relate to tangent, we notice that if both cosine and sine are positive, then cotangent would be positive; if one is negative, then cotangent is negative; if both are negative, then cotangent is positive. Unlike cosine and sine, cotangent graphs also have asymptotes Cotangent graphs are identically related to cosine and sine as its reciprocal, but have different asymptotes. If sine (y) were to be equal to 0, that would result in an asymptote. If we were to look at a unit circle, we should know that the y value would equal 0 at 0 degrees and 180 degrees. Converting back to the cotangent graph and radians, we know that our asymptotes would be at 0 and pi.
3. Secant
Unlike tangent and cotangent, secant is only related to the cosine graph. The unit circle ratio of secant is 1/cosine. If cosine were to have positive values on its graph, so would secant, and vice versa. Because secant is the reciprocal of cosine, we notice that if we were to have a low value on the cosine graph, the reciprocal of that value would be bigger on the secant graph, and vice versa. We should also notice that each curve of the secant graphs touch the "mountains" and "valleys" of the cosine graph. Looking at a graph, if we were to have a value of 0 on the cosine graph (0/1), we should know that that is where our asymptotes would be for the secant graph(1/0 = undefined).
4. Cosecant
Again, unlike tangent and cotangent, cosecant is only related to the sine graph. The unit circle ratio of cosecant is 1/sine. If sine were to have positive values on its graph, so would cosecant, and vice versa. Because cosecant is the reciprocal of sine, we notice that if we were to have a low value on the sine graph, the reciprocal of that value would be bigger on the cosecant graph, and vice versa. We should also notice that each curve of the cosecant graphs touch the "mountains" and "valleys" of the sine graph. Looking at a graph, if we were to have a value of 0 on the sine graph (0/1), we should know that that is where our asymptotes would be for the cosecant graph(1/0 = undefined).
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