Solving for sin 2x Given sin x cos x 2/3: Two Different Methods

Given that (sin x cos x frac{2}{3}), the objective here is to find the value of (sin 2x). This can be achieved using two different methods. Let's go through each step by step.

Method 1

Starting with the given equation:

(sin x cos x frac{2}{3})

First, let's square both sides of the equation:

(sin^2 x cos^2 x left(frac{2}{3}right)^2 frac{4}{9})

Using the Pythagorean identity, (sin^2 x cos^2 x 1), we can express (sin^2 x) and (cos^2 x) as follows:

(1 - 2 sin x cos x sin^2 x cos^2 x - 2 sin x cos x)

Substituting the known value:

(1 - 2 cdot frac{2}{3} 1 - frac{4}{3} -frac{1}{3})

Since (sin 2x 2 sin x cos x), we can find:

(sin 2x 2 cdot frac{2}{3} frac{4}{3} - 1 -frac{5}{9})

Method 2

Another approach involves a more direct manipulation of the given equation:

(sin x cos x frac{2}{3})

Multiplying both sides by (frac{1}{sqrt{2}}), we get:

(sin frac{pi}{4} sin x cos frac{pi}{4} cos x frac{sqrt{2}}{3})

Using the product-to-sum identities, we can rewrite this as:

(sin frac{pi}{4} sin x cos frac{pi}{4} cos x sin x cos x frac{1}{2} (sin (frac{pi}{4} x) sin (frac{pi}{4} - x)) frac{sqrt{2}}{3})

Simplifying further by letting:

(cos (frac{pi}{4} - x) frac{sqrt{2}}{3})

Thus,

(x frac{pi}{4} - cos^{-1} frac{sqrt{2}}{3})

Substituting back into the expression for (sin 2x) using trigonometric identities and simplifying, we find:

(sin 2x -frac{5}{9})

Conclusion

In both methods, we have determined that (sin 2x -frac{5}{9}). By employing different trigonometric identities and identities for sine and cosine, we can solve such problems effectively. This demonstrates the power of manipulating trigonometric equations using various identities and properties.

Understanding these methods is crucial for solving a wide range of trigonometric problems and is particularly important for students and professionals in mathematics, physics, and engineering. Practice with these techniques will greatly enhance your problem-solving skills in these fields.