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Chapter 11: Problem 151

$$ \cos ^{2} a+\cos ^{2}\left(a+120^{\circ}\right)+\cos ^{2}\left(a-120^{\circ}\right)=\frac{3}{2} $$

Short Answer

Expert verified
The solutions for the trigonometric equation are \( a = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

Step by step solution

01

Use the Trigonometric Identity

Expand each term using the cos identity \( \cos(120^o) = -\frac{1}{2} \). Thus, the equation becomes \( \cos^2 a + \cos^2(a - \frac{1}{2}) + \cos^2(a + \frac{1}{2}) = \frac{3}{2} \)
02

Simplify the Equation

Substitute \( \cos^2 x = 1 - \sin^2 x \) into the equation and get three equations: \( \cos^2 a , \(1 - \sin^2(a - \frac{1}{2}), and \( 1 - \sin^2(a + \frac{1}{2}) = \frac{3}{2} \)
03

Solve the Equation

Add up all three equations and solve. This gives the final solution as \( a = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

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