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I know that there can't be two right angles in a right-angled triangle. But, I have seen many proofs that sin 90° = 1.

The proofs make sense analytically, but how do we know that sin(90°) (or, for that matter, sin 0° = 0) is constant as is sin ɸ such that 0° < ɸ < 90°, i.e., in the latter, we know the ratio of opposite side to the hypotenuse is fixed for a given angle but how can this be known in the above case

I have seen the unit circle animations, but historically, before extending trigonometry to the unit circle in the Cartesian plane, we must have figured this out using the standard right triangle definition of trig functions. Please do give an explanation in the right triangle definition.

MarianD
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Sathvik R.
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    This answer of mine may (or may not) be helpful. – Blue Jan 27 '21 at 14:07
  • Well, what happens as the angle tends to 90 degrees? – Joshua Wang Jan 27 '21 at 14:07
  • @Blue My question actually stems from your answer. What seems to confuse me is how you could say Sin(90°) = 1. I mean, how could you say that the ratio will be constant? – Sathvik R. Jan 27 '21 at 14:14
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    Per the comment of @JoshuaWang, in Analytical Geometry (or Trigonometry) the sine function represents the ratio of $$\frac{\text{opposite}}{\text{hypotenuse}}.$$ As the angle approaches $90^\circ$, the length of the opposite leg approaches the length of the hypotenuse. – user2661923 Jan 27 '21 at 14:17
  • @JoshuaWang Well, it makes sense when expressed with limits, but is there any way of representing this in terms of the ratio? – Sathvik R. Jan 27 '21 at 14:17
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    @SathvikR.: "What seems to confuse me is [...] how could you say that the ratio will be constant?" It's okay to find my reasoning in that other answer unconvincing. The evidence is admittedly circumstantial ... and, yet ... there are so many circumstances, it just seems like $\sin 90^\circ$ wants to be $1$! Cognitive tension about this is natural; it's what motivates a mathematician to seek a broader context in which such tension simply vanishes. The unit circle (re-)definition of sine is one such context; the power series is another. – Blue Jan 27 '21 at 15:18
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    You have begun your question with "I know that there can't be two right angles in a right-angled triangle." and that is your answer. The right triangle definition of trig functions only makes sense for angles strictly between $0^\circ$ and $90^\circ$. – Somos Jan 27 '21 at 16:59
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    @Blue Wow, truly fascinating. – Sathvik R. Jan 28 '21 at 04:56

1 Answers1

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I assume by right triangle definitions of trig you mean, the rule that with length $1$ hypotenuse:

$$\sin(\theta) = A$$

Where A is the side opposite of theta.

If you use the law of sines on the right triangle you get that

$$\frac{\sin(90)}{H} = \frac{\sin(\theta)}{A}.$$

Plugging in the values we know:

$$\frac{\sin(90)}{1} = \frac{A}{A} = 1.$$

Hopefully I followed all of the necessary restrictions.

Xander Henderson
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open problem
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