Solving Trigonometric Expression with Given Sine Value

The question asks for the value of the expression $ (3\sin A - 4\cos A)^2 + (4\sin A + 3\cos A)^2 $ given that $ \sin A = \frac{2}{3} $. We can solve this by expanding the terms and simplifying using trigonometric identities.

Step 1: Expand the first term

Expand $ (3\sin A - 4\cos A)^2 $: $ (3\sin A - 4\cos A)^2 = (3\sin A)^2 - 2(3\sin A)(4\cos A) + (4\cos A)^2 $ $ = 9\sin^2 A - 24\sin A \cos A + 16\cos^2 A $

Step 2: Expand the second term

Expand $ (4\sin A + 3\cos A)^2 $: $ (4\sin A + 3\cos A)^2 = (4\sin A)^2 + 2(4\sin A)(3\cos A) + (3\cos A)^2 $ $ = 16\sin^2 A + 24\sin A \cos A + 9\cos^2 A $

Step 3: Add the expanded terms

Now, add the results from Step 1 and Step 2: $ (9\sin^2 A - 24\sin A \cos A + 16\cos^2 A) + (16\sin^2 A + 24\sin A \cos A + 9\cos^2 A) $

Step 4: Simplify the combined expression

Combine like terms. Notice that the $ \sin A \cos A $ terms cancel out: $ = (9\sin^2 A + 16\sin^2 A) + (16\cos^2 A + 9\cos^2 A) + (- 24\sin A \cos A + 24\sin A \cos A) $ $ = 25\sin^2 A + 25\cos^2 A + 0 $

Step 5: Apply the Pythagorean Identity

Factor out the common coefficient, 25: $ = 25(\sin^2 A + \cos^2 A) $ Using the fundamental trigonometric identity $ \sin^2 A + \cos^2 A = 1 $: $ = 25(1) $ $ = 25 $

Conclusion

The value of the expression $ (3\sin A - 4\cos A)^2 + (4\sin A + 3\cos A)^2 $ is 25. Note that this result is independent of the specific value of $ \sin A $ (in this case, $ \frac{2}{3} $), as the expression simplifies to a constant value.