Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.
Sine and cosine do not have asymptotes because they can't be undefined. In order for a trig function to have an asymptote, it has to be undefined, meaning the denominator has to be zero. Sine is y/r and cosine is x/r. We know from the Unit Circle that the radius (r) is always one; therefore, their denominators can never be undefined.
The other four trig graphs, on the other hand, do have asymptotes. There, however, are restrictions. Cosecant (r/y) and cotangent (x/y) have the same denominator: y. On the unit circle, "y" can be zero at 0º (1,0) and 180º (-1,0). As a result, they can still have asymptotes but they can also have no asymptotes as well. Another example is secant (r/x) and tangent (y/x). They also have the same denominator: x. "X" can be zero at 90º (0,1) and 270º (0,-1). Whenever x is zero, it will have an asymptote. When it's not zero, it won't have asymptotes. These four trig graphs do have asymptotes but there are also times where they don't have asymptotes like sine and cosine.
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