A Proof without Calculus

Now, solve \sin(x)=x.

By inspection, x=0.

Is this the only solution?

Visually, it is difficult to tell (see Fig. 1 and Fig. 2)

Fig. 1

Fig. 2

However, we can prove that 0 is the only solution:

Fig. 3

If 0 < x \leq 1 < \frac{\pi}{2} then from Fig. 3, we have

Area of triangle OAB < Area of circular sector OAB.

That is,

\frac{1}{2}\cdot 1 \cdot \sin(x)< \frac{1}{2}\cdot 1 \cdot x.


\forall 0< x \leq 1, \sin(x) < x\quad\quad\quad(1)

If x>1 then

\sin(x) \leq 1 \overset{x>1}{\implies} \sin(x) \leq 1< x \implies \forall x>1, \sin(x) <x\quad\quad\quad(2)

Put (1) and (2) together, we have

\forall x > 0, \sin(x) < x\quad\quad\quad(3)

If x < 0 then

-x > 0 \overset{(3)}\implies \sin(-x) < -x \overset{\sin(-x)=-\sin(x)}\implies -\sin(x) < -x \implies \sin(x) > x


\forall x < 0, \sin(x) > x\quad\quad\quad(4)

Therefore by (3) and (4), we conclude that

only when x = 0, \sin(x)=x.


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