Exploring Trigonometric Identities and Finding the Value of ( sin x - cos x )

In this article, we will explore the application of trigonometric identities to solve a specific equation and determine the value of ( sin x - cos x ). The equation in question is ( sin x cos x frac{1}{2} ).

Solving the Given Equation

The given equation is ( sin x cos x frac{1}{2} ). We can use the double-angle identity to simplify and solve this problem. The double-angle identity for sine is:

(sin x cos x frac{1}{2} sin 2x)

Using this identity, we can rewrite the given equation as:

( frac{1}{2} sin 2x frac{1}{2} )

This simplifies to:

( sin 2x 1 )

To find the general solution for ( sin 2x 1 ), we solve:

( 2x frac{pi}{2} 2kpi quad text{for } k in mathbb{Z} )

Solving for ( x ) gives:

( x frac{pi}{4} kpi quad text{for } k in mathbb{Z} )

Calculating ( sin x - cos x )

Now, we need to determine the value of ( sin x - cos x ). We will use the solutions for ( x ) to find the values of ( sin x ) and ( cos x ).

For ( k 0 )

Let's take ( x frac{pi}{4} ).

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

Thus:

( sin frac{pi}{4} - cos frac{pi}{4} frac{sqrt{2}}{2} - frac{sqrt{2}}{2} 0 )

For ( k 1 )

Let's take ( x frac{5pi}{4} pi frac{pi}{4} ).

( sin frac{5pi}{4} -frac{sqrt{2}}{2}, quad cos frac{5pi}{4} -frac{sqrt{2}}{2} )

Thus:

( sin frac{5pi}{4} - cos frac{5pi}{4} -frac{sqrt{2}}{2} - left(-frac{sqrt{2}}{2}right) 0 )

In both cases, we find that:

( sin x - cos x 0 )

Using Other Methods to Confirm the Solution

Here are a few alternative methods to confirm that ( sin x - cos x 0 ) given ( sin x cos x frac{1}{2} ).

Method 1: Using Squaring and Simplifying

( sin x - cos x^2 sin^2 x - 2 sin x cos x cos^2 x ) ( sin^2 x - cos^2 x - 2 sin x cos x ) ( 1 - 2 cdot frac{1}{2} 1 - 1 0 )

Thus, ( sin x - cos x 0 ).

Method 2: Using the Double-Angle Identity

( sin x cos x frac{1}{2} Rightarrow 2 sin x cos x 1 )

( sin^2 x - cos^2 x 2 sin x cos x - 2 sin x cos x 1 - 2 cdot frac{1}{2} 1 - 1 0 )

From these methods, we confirm that: