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Today I explained to my students how the value of $\sin(90°)$ is 1 by first drawing a right angled triangle and taking $90°$ as a reference angle, and told them that now $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Then I used pythagorean relation to show that $\cos(90°)$ is 0, i.e $\sin^2(\theta)=1-\cos^2(\theta)$. And finally I used quotient relation to show how $\tan(90°)$ is infinity. The students didn't seem much convinced by my methods as they were asking why we couldn't find the value of $\cos(90°)$ and $\tan(90°)$ the same way we did to find the value of $\sin(90°)$. Is there any other ideas to show the value of $\sin(90°)$, $\cos(90°)$ and $\tan(90°)$ understandble to them?

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    $\begingroup$ "to show how $\tan(90^\circ)$ is infinity" Technically, this is incorrect. Although the limit of $\tan\theta$ as $\theta$ approaches $90^\circ$ from below is infinity, what is the limit as $\theta$ approaches $90^\circ$ from above? $\endgroup$ – Joel Reyes Noche Jun 1 '14 at 23:46
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Yes: Use the definitions of these functions on the unit circle. All of these facts then become geometrically apparent. These concepts do not even really apply to right triangles, since you would need a triangle with 2 right angles, which is not possible.

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I don't know if, in good faith, you can really assert than $\sin(90^\circ)=1$ without your students having a very firm grasp of the unit circle or at least some kind of limiting process. But, then, if the students have a firm grasp of the unit circle, your argument for why $\cos(90^\circ)=0$ should be seen as unnecessarily complicated. And then, as mentioned above, you shouldn't say that $\tan(90^\circ)$ is infinite but you should really say that it's undefined (careful, though, if a curious student enters $\tan(\pi/2)$ into a certain free online calculator, they'll get something like $1.63317787 \times 10^{16}$).

What you could do to accomplish what it seems you want to accomplish (i.e., to calculate values of sine and cosine using identities) you could have them do a bunch of consistency checks in the form of exercises such as:

  • Using the double angle prove that $\sin(90^\circ)=1$
  • Using the Pythagorean identity, prove $\cos(90^\circ)=0$.
  • By evaluating $\tan(x)$ using the definition at $x=89^\circ,\, 89.9^\circ,\, 89.99^\circ,\dots$ and at $x=91^\circ,\, 90.1^\circ,\,90.01^\circ,\dots$ try to explain a little more about they sense in which $\tan(90^\circ)$ is undefined. Now try to explain it using the unit circle.
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A mix of your approach, Steven's answer, and also the graphed Sine, Cosine, and Tangent functions. To Joel's point, the graph will clearly show tan θ going to either plus or minus infinity at 90∘ depending on which side you approach from. While sine and cosine run into issues on right triangles, the curves are easy to understand and have no discontinuities.

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