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I just wanted sin(x) in terms of algebraic operations like addition,multiplication,etc. And can we extract that from the right angled triangle sides ratio definition, sine=altitude/hypotenuse
or we have to involve calculus.

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If an infinite polynomial is fine with you, we can use the Taylor Series for $\sin(x)$. It is derived though calculus, however.

$$\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$$

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