Problem 151
Question
$$ \cos ^{2} a+\cos ^{2}\left(a+120^{\circ}\right)+\cos ^{2}\left(a-120^{\circ}\right)=\frac{3}{2} $$
Step-by-Step Solution
Verified Answer
The solutions for the trigonometric equation are \( a = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)
1Step 1: Use the Trigonometric Identity
Expand each term using the cos identity \( \cos(120^o) = -\frac{1}{2} \). Thus, the equation becomes \( \cos^2 a + \cos^2(a - \frac{1}{2}) + \cos^2(a + \frac{1}{2}) = \frac{3}{2} \)
2Step 2: 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 \( 1 - \sin^2(a + \frac{1}{2}) = \frac{3}{2} \)
3Step 3: Solve the Equation
Add up all three equations and solve. This gives the final solution as \( a = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)
Other exercises in this chapter
Problem 149
$$ \tan 3 A \tan \angle A \tan A=\tan 3 A-\tan \angle A-\tan A $$
View solution Problem 150
$$ \sqrt{2+\sqrt{2+2 \cos 4 \theta}}=2 \cos \theta $$
View solution Problem 152
$$ \cos ^{2} a+\cos ^{2}\left(a+120^{\circ}\right)+\cos ^{2}\left(a-120^{\circ}\right)=\frac{3}{2} $$
View solution Problem 153
$$ (\tan 4 A+\tan 2 A)\left(1-\tan ^{2} 3 A \tan ^{2} A\right)=2 \tan 3 A \sec ^{2} A $$
View solution