SIN 60° – VALUE AND DERIVATION

The trigonometric ratios measure the ratio of the lengths of the sides of the triangle. We have six such trigonometric ratios namely sin, cosec, cos, sec, tan and cot. We will discuss sin 60° in this article. Now, let us take a look at the right-angled triangle.

Here, the longest side c denotes the hypotenuse, a and b denote the other two sides of the right-angled triangle. The side opposite the angle Ɵ is the perpendicular which is side a and b is the base. 

 The following table shows how to calculate the value of the six different ratios:

DERIVATION OF SIN 60°

Let’s take an equilateral triangle with each angle as 60°. Now, draw a perpendicular to the line BC. The length of the sides of the triangle is ‘a’ units. The perpendicular divides the line into half at D, thus making BD = DC = a/2.

Now, we use Pythagoras theorem to find the length of the perpendicular AD.

    AD^2+BD^2=AB^2  

⇒ AD^2+(a/2)^2=a^2  

⇒ AD^2=a^2-(a/2)^2  

⇒ AD^2=(4a^2-a^2)/4  

⇒ AD^2=3a^2/4  

∴ AD=\sqrt{3}a/2  

Now,

Sin 60° = side opposite the angle/ hypotenuse

Sin 60° = (\sqrt{3}a/2)/a

⇒ Sin 60° = \sqrt{3}/2

So, we know from this derivation that sin 60° = \sqrt{3}/2.

TRIGONOMETRIC RATIOS AND THEIR VALUES

Trigonometric ratios can be calculated for any angle but there are certain standard angles whose values are shared in the table below.

The table also shows the value of sin 60° = \sqrt{3}/2.