What is the Value of Cos 30 Degrees ?

Solution :

The value of cos 30 degrees is \(\sqrt{3}\over 2\).

Proof :

Consider an equilateral triangle ABC with each side of length of 2a. Each angle of \(\Delta\) ABC is of 60 degrees. Let AD be the perpendicular from A on BC.

\(\therefore\)   AD is the bisector of \(\angle\) A and D is the mid-point of BC.equilateral triangle

\(\therefore\)   BD = DC = a and \(\angle\) BAD = 30 degrees.

In \(\Delta\) ADB, \(\angle\) D is a right angle, AB = 2a and BD = a

By Pythagoras theorem,

\(AB^2\) = \(AD^2\) + \(BD^2\)  \(\implies\)  \(2a^2\) = \(AD^2\) + \(a^2\)

\(\implies\)  \(AD^2\) = \(4a^2\) – \(a^2\) = \(3a^2\)   \(\implies\)  AD = \(\sqrt{3}a\)

Now, In triangle ADB, \(\angle\) BAD = 30 degrees

By using trigonometric formulas,

\(cos 30^{\circ}\) = \(base\over hypotenuse\) = \(b\over h\)

\(cos 30^{\circ}\) = side adjacent to 30 degrees/hypotenuse = \(AD\over AB\) = \(\sqrt{3}\over 2a\) = \(\sqrt{3}\over 2\)

Hence, the value of \(cos 30^{\circ}\) = \(\sqrt{3}\over 2\)

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