Question

How do you find the exact values of \[\cos 75\] degrees using the half angle formulae ?

Answer 1

Hint: Half angle formulae is the method in which we find the value for the double angle as given and then after simplification using the trigonometric properties we have to get the value for the required angle, this method is suitable for the half angles of the angles whose value are known to us, for example we know the values for forty five degree then solving for its half is very easy.

Formulae Used:
\[\cos 2t = 2{\cos ^2}t - 1\]

Complete step by step solution:
The given question is \[\cos 75\]
Here we have to solve this expression by using the half angle method, so for that we have to assume the given angle as a variable say “x”, now on solving we get:
\[\Rightarrow \cos 75 \\
\Rightarrow \cos 75 = \cos t \\ \]
For half angle method lets double the angle we get:
\[\Rightarrow \cos 2t = \cos 150 \\
\Rightarrow \cos 150 = \cos (180 - 30) = - \cos 30 = - \dfrac{{\sqrt 3 }}{2} \\ \]
Now by using the trigonometric property that is:
\[ \Rightarrow \cos 2t = 2{\cos ^2}t - 1\]
Using this and applying for our equation we get:
\[\Rightarrow \cos 2t = - \dfrac{{\sqrt 3 }}{2} = 2{\cos ^2}t - 1 \\
\Rightarrow 2{\cos ^2}t - 1 = - \dfrac{{\sqrt 3 }}{2} \\
\Rightarrow 2{\cos ^2}t = - \dfrac{{\sqrt 3 }}{2} + 1 = \dfrac{{2 + \sqrt 3 }}{2} \\
\Rightarrow {\cos ^2}t = \dfrac{{2 + \sqrt 3 }}{4} \\
\Rightarrow \cos t = \sqrt {\dfrac{{2 + \sqrt 3 }}{4}} = \dfrac{{\sqrt {2 + \sqrt 3 } }}{2} \\
\therefore \cos 75 = \dfrac{{\sqrt {2 + \sqrt 3 } }}{2} \\ \]
This is the required value for the given angle for the trigonometric function.

Note: For finding the angles using double angle method you have to solve by the certain regular steps, without that you cannot get the required answer, these steps are standard and should be followed, only the property used in every question will not be same, it depends on the function what is asked for, and accordingly best suited property should be applied for getting the answer. Here the property used is required by the question, by other properties it might take some more steps to solve.