Trigonometric Ratio: Sin 0° – value and derivation

SINE (SIN) 0°

In mathematics, we study trigonometry which deals with the sides and angles of triangles. We can evaluate angles and sides using the various trigonometric ratios. These are the ratios of two sides of a right-angled triangle (90° angle). Sine, cosine, tangent, cosecant, secant and cotangent – these are the 6 trigonometric ratios.

We will learn about the value of Sin 0° in this article.

A right-angled triangle has two sides and a hypotenuse. The longest side of a right-angled triangle is the hypotenuse.

Sin is the ratio of the side opposite to the angle and the hypotenuse of the right-angled triangle.

Sin Ɵ = side opposite to the angle Ɵ/ hypotenuse

Let side c be the hypotenuse. The measure of the angle between a and b is 90°. Since we measure the side opposite the angle Ɵ (perpendicular), here

Sin Ɵ = \frac {a}{c}

In a right-angled triangle, it is possible to find the values of an unknown side if the value of the other two sides is given. This is done using the Pythagoras theorem. According to this theorem,

And here, 

perpendicular^2+base^2=hypotenuse^2

a²+b²=c²

Now, the other ratios are also combinations of different sides of a triangle. In the table below, you will see how to find the values of all the six trigonometric ratios:

There are standard angles for which the values of the trigonometric ratios are well known. Given below is the trigonometric table for standard angles:

As we can see, the value of sin 0° is 0.

DERIVATION OF THE VALUE OF SIN 0°

Basically, sine is the ratio between the side opposite the angle and the hypotenuse so if we consider sin 0° which means that the angle between the hypotenuse and its adjacent side (which is base here) is 0°. It is only possible when the hypotenuse coincides with the adjacent side thus making the opposite side equal to 0. Thus the ratio becomes 0/a since RQ (the hypotenuse) coincides with the side PQ. 

Therefore, the value of sin 0° is 0.