First slide

Multiple and sub- multiple Angles

Question

$If \cos A + \cos B + \cos C = 0 and \cos 3 A + \cos 3 B + \cos 3 C = λ \cos A \cos B \cos C then λ is$

Moderate

A

8
B

10
C

12
D

16

Solution

$\begin{array}{l} If \cos A + \cos B + \cos C = 0 then \cos^{3} A + \cos^{3} B + \cos^{3} C = 3 \cos A \cos B \cos C \\ Given \cos 3 A + \cos 3 B + \cos 3 c = λ \cos A \cos B \cos C \\ (4 \cos^{3} A - 3 \cos A) + (4 \cos^{3} B - 3 \cos B) + (4 \cos^{3} C - 3 \cos C) \\ = λ \cos A \cos B \cos C = 4 (\cos^{3} A + \cos^{3} B + \cos^{3} C) - 3 (\cos A + \cos B + \cos C) \\ = 4 (3 \cos A \cos B \cos C) - 3 (0) = 12 \cos A \cos B \cos C \end{array}$

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