Question

If $\cos \theta = (\dfrac{1}{2})(a + \dfrac{1}{a})$ then the value of $\cos 3\theta $ is
A. $(\dfrac{1}{8})(a^3 + \dfrac{1}{a^3})$
B. $(\dfrac{3}{2})(a + \dfrac{1}{a})$
C. $(\dfrac{1}{2})(a^3 + \dfrac{1}{a^3})$
D. $(\dfrac{1}{3})(a^3 + \dfrac{1}{a^3})$

Answer 1

Hint: Before we proceed to solve the problem, it is important to know about the trigonometric formula to be used. To solve this question, you could directly use the trigonometric identities. $\cos 3\theta $ is an identity in trigonometry and can be expressed in terms of the $\cos \theta $.

Formula Used:
$\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta $
${x^3} + {y^3} = (x + y)({x^2} + {y^2} - xy)$

Complete step by step Solution:
We need to find the value of $\cos 3\theta $. This is the multiple-angle formula.

Given that
$\cos \theta = (\dfrac{1}{2})(a + \dfrac{1}{a})$
$\cos 3\theta = \cos \theta (4{\cos ^2}\theta - 3)$
$ \Rightarrow \cos 3\theta = (\dfrac{1}{2})(a + \dfrac{1}{a})[4 \times {\{ \dfrac{1}{2}(a + \dfrac{1}{a})\} ^2} - 3]$
$ \Rightarrow \cos 3\theta = (\dfrac{1}{2})(a + \dfrac{1}{a})[\dfrac{4}{4}({a^2} + \dfrac{1}{{{a^2}}} + 2) - 3]$
$ \Rightarrow \cos 3\theta = (\dfrac{1}{2})(a + \dfrac{1}{a})({a^2} + \dfrac{1}{{{a^2}}} - 1]$
$ \Rightarrow \cos 3\theta = (\dfrac{1}{2})({a^3} + \dfrac{1}{{{a^3}}})$ [since ${x^3} + {y^3} = (x + y)({x^2} + {y^2} - xy)$]
Therefore, correct option is C

Note:We can also solve this question by using the double angle formula of cos and then using the trigonometric identities. To solve these types of questions in an untedius way you should know the triple angle formula for both cosine and sine angle. You can solve fastly if you know the formula correctly.