Trigonometric function sine and angle of the triangle often appears in rational form when the angle tends to zero in limits. The result of it is used as a formula in dealing the functions in which trigonometrical functions involve in limits.

Consider a triangle, and the angle of triangle is assumed $x$. Therefore, the trigonometric function sine at angle is expressed as $\mathrm{sin}x$ mathematically.

Assume, the angle between adjacent side and hypotenuse is very small which means the value of angle is approximately zero. In such a case, the limit of ratio of sine function to angle when an angle tends to zero, is expressed in mathematics as follows.

$\underset{x\to 0}{lim}\⁡\frac{sinx}{x}$

Here, the meaning $x\to 0$ is, the value of angle $x$ is approximately zero.

Consider some angles whose value is approximately zero for understanding the difference between angles and associated sine values. Input each angle to sine function and observe the value of sine function. Note, all angles are input in radians scale.

In first example, an angle is considered $x=0.000056$ radians. Substitute the angle in sine function and you get the associated value.

$x=0.000056$ rad $\Rightarrow sin0.000056=0.0000559999\approx 0.000056$

You perceive that the output value of sine function is almost equal to the submitted angle. Also observe following examples.

$x=0.000235$ rad $\Rightarrow sin0.000235=0.0002349999\approx 0.000235$

$x=0.0045$ rad $\Rightarrow sin0.0045=0.0044999848\approx 0.0045$

$x=0.089546$ rad $\Rightarrow sin0.089546=0.089426377\approx 0.089546$

$x=0.256489$ rad $\Rightarrow sin0.256489=0.253685979\approx 0.256489$

$x=0$ rad $\Rightarrow sin0=0$

In all the above examples, the output value of sine function is same as the angle even in the case of angle is equal to zero. Remember, sine function returns same value or approximate value if the angle is approximately zero. Come back to our mathematical proof.

$\underset{x\to 0}{lim}\⁡\frac{sinx}{x}$

In this limits equation, you can perceive that the value of angle is reaches the zero $(x\to 0)$. Therefore, sine function must return the value same as the angle. For this reason, $sinx\approx x$. So, you can replace $sinx$ with $x$ in limits equation.

$\Rightarrow \underset{x\to 0}{lim}ApplyFunction;\frac{x}{x}$

The values are same in both numerator and dominator of the equation. So, they both get cancelled mathematically.

$\Rightarrow \underset{x\to 0}{lim}ApplyFunction;1$

In the limits equation, there is no $x$. Thence, the value of this limits function is the remaining constant. Here, the constant is one.

$\Rightarrow \underset{x\to 0}{lim}ApplyFunction;1=1$

Therefore, it is proved that limit of the ratio of sine function at an angle to angle is one when the value of angle of a triangle is approximately zero.

$\underset{x\to 0}{lim}\⁡\frac{sinx}{x}=1$

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