Cos is the abbreviation for the trigonometric function ‘Cosine’. Cosine is defined as the ratio of the length of the base of a right-angle triangle to its hypotenuse (the longest and slant side). Cos (x) = base/hypotenuse or b/h, where x is the angle between the two sides.
The angle extended by the base and the hypotenuse is the angle we associate with cos. Cos 30 degrees is defined as an angle of 30 degrees that lies in between the base and hypotenuse of a right-angle triangle. The value of cos 30 is predefined as √3/2 (in fractional form), which is equivalent to sin 60 degrees. The reciprocal of cos is ‘secant’ or ‘sec’.
There are three rudimentary values for cos 30 degrees. In trigonometric ratio, as discussed above, the value is in fractional form equal to √3/2. In the circular system, using trigonometric functions in the cartesian plane, we get that the value of cos 30 is pi/6 or 180/6. This is although the particular solution to one cycle if n number of cycles are present then the value becomes (n x pi / 6). The last approach for determining the value of cos 30 is the centesimal system which gives the value as Cos 33 (1/3) g. Irrespective of the method, the value of cos 30 in decimal is always equal to 0.8660254037.
There are two ways determined by mathematicians to prove the value of cos 30, namely the theoretical approach and the practical approach. Let us understand both concepts in detail.
Let us consider a triangle KLM such that it is a right-angled triangle with base KL, height LM height, and KM hypotenuse. Let us assume that the length of the hypotenuse is ‘a’ unit. Also, let us consider the angle between the base and hypotenuse is 30 degrees. There is a property that states that “For every right-angle triangle having 30 degrees angle the opposite side of the triangle is half the length of the hypotenuse.” Therefore, LM = ‘a/2’. Let us now determine the base of the triangle using the Pythagoras theorem.
According to the Pythagoras Theorem- Hypotenuse^{2} = Base^{2} + Height^{2}
In triangle KLM,
KM ^{2} = KL^{2} + LM^{2}
Hypotenuse = a
Opposite side = a/2
a^{2} = (a/2)^{2} + KL^{2}
a^{2} = a^{2}/4 + KL^{2}
a^{2} – a^{2}/4 = KL^{2}
(4a^{2} – a^{2})/4 = KL^{2}
3a^{2}/4 = KL^{2}
√ (3) * a/2 = KL
KL/a = √ (3) / 2
Since KL is the adjacent side of the triangle
Cos 30 = √3 / 2 = 0.8660254037
To practically prove the value of cos 30, we must draw a right-angle triangle with one angle equivalent to 30 degrees from scratch and then find the ratio of the lengths of the hypotenuse and the base. The steps are mentioned as follows: