cos²A + cos² (60° + A) + cos² (60° - A)=3/2
Answers
I think there should be a cos²A in right hand side. Then it will be full filled. I hope it will help you.
Therefore the given trigonometric function cos²A + cos² (60° + A) + cos² (60° - A)=3/2 is true for every value of 'A'.
Given:
The trigonometric equation: cos²A + cos² (60° + A) + cos² (60° - A)=3/2
To Find:
The value of 'A'.
Solution:
The given question can be solved very easily as shown below.
Given trigonometric function: cos²A + cos² (60° + A) + cos² (60° - A)=3/2
We know that, cos² x = ( 1 + cos 2x )/2
⇒ ( 1 + cos 2A )/2 + ( 1 + cos 2 (60 + A )/2 ) + ( 1 + cos 2(60 - A)/2 ) = 3/2
⇒ ( 1 + cos 2A) + ( 1 + cos(120 + 2A ) ) + ( 1 + cos (120 - 2A ) = 3
→ We know that Cos ( x + y ) = cos x. cos y - sin x. siny
→ And also cos ( x - y ) = cos x. cos y + sin x. sin y
⇒ Now, 3 + cos 2A + cos 120. cos 2A - sin 120. sin 2A + cos 120. cos 2A + sin 120. sin 2A = 3
⇒ cos 2A + 2. cos 120. cos 2A = 0
⇒ cos 2A [ 1 + 2cos 120 ] = 0
⇒ cos 2A [ 1 + 2 ( -1/2 ) ] = 0
⇒ cos 2A × 0 = 0
So the relation holds for every value of 'A'.
Therefore the given trigonometric function cos²A + cos² (60° + A) + cos² (60° - A)=3/2 is true for every value of 'A'.
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