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I'm just curious, considering how similar the graphs of the sine & cosine functions are in shape, why is the shape of the tangent's graph so different, despite being used in very similar types of problem?


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up vote 18 down vote accepted

The sine and cosine are related to the two legs of a right triangle; they are the ratio of the leg opposite the angle to the hypotenuse and the ratio of the leg adjacent to the angle to the hypotenuse, respectively. Since the hypotenuse is always longer than the two legs, this ratio is always less than $1$. Moreover, since the angles adjacent to a leg and opposite a leg are related by $\alpha = \pi/2 - \beta$, the sine and cosine necessarily have similar forms. By contrast, the tangent is the ratio of the legs to each other, and this can take on any value and goes to infinity when one of the legs goes to zero. [Of course this entire account glosses over negative values, but you can do the same thing more rigorously by instead talking about the Cartesian coordinates of points on the unit circle.]

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Why the graph of $x/x^2$ is different from $x$ and $x^2$?

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Oh, now I see! Thanks. – Danny King Mar 26 '11 at 15:58

Take a look at the Unit Circle. This graphic shows where all the trigonometric functions come from.

Now take a look at an animated version of sine and tangent.Animated tangent

I can't take any credit for these, but they may be helpful in visualize how the functions arise.

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Those animations are wonderful! – Danny King Mar 27 '11 at 11:20
@Ryan That is so awesome! – Pedro Tamaroff Feb 16 '12 at 3:15

These pictures may also shed some light:

(Maybe someone who knows about formatting can do some further editing.)

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