Question

If \((\sin x+\cos x)^{2}=k,\) then what is the value of \(\sin 2 x\) in terms of \(k ?\)

Short answer

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Step-by-step solution

Expand the given equation

Start by expanding \((\sin x + \cos x)^{2} = k\). The expanded form is: \((\sin x + \cos x)^2 = \sin^2 x + 2 \sin x \cos x + \cos^2 x\).

Use Pythagorean identity

Replace \(\sin^2 x + \cos^2 x\) with 1 using the Pythagorean identity. So, we now have: \1 + 2 \sin x \cos x = k\.

Key concepts

Pythagorean identity

In trigonometry, the Pythagorean identity is a fundamental relation between the sine and cosine of an angle. It states that for any angle x, the following equation holds true: \(\text{sin}^2 x + \text{cos}^2 x = 1 \). This identity is incredibly useful when simplifying expressions. For instance, in our current problem, we used it to replace \(\text{sin}^2 x + \text{cos}^2 x \) with 1. Without this identity, solving trigonometric equations would be much more complex. So, always remember that this identity can simplify your calculations significantly.

trigonometric expressions

Trigonometric expressions include a mix of trigonometric functions such as sine (sin), cosine (cos), and tangent (tan). Manipulating these expressions often involves using various identities and formulas. In our exercise, we had \((\text{sin} x + \text{cos} x)^2 = k\). To simplify or solve such expressions, we often expand, combine like terms, or use identities. Understanding how to work with these functions and their properties is key to solving trigonometric problems effectively.

sin 2x formula

The formula for the double angle of sine is an essential trigonometric identity: \(\text{sin} 2x = 2 \text{sin} x \text{cos} x\). This tells us that the sine of twice an angle is twice the product of the sine and cosine of the original angle. In our problem, we identified \(\text{sin} 2x\) as \((2 \text{sin} x \text{cos} x)\), which is directly related to the expanded form of \((\text{sin} x + \text{cos} x)^2\). By understanding this formula, we could make the connection between the given equation and the value of \(\text{sin} 2x\).

expanding equations

Expanding equations involves multiplying out expressions to simplify and solve them. In the given example, we started with \((\text{sin} x + \text{cos} x)^2 = k\). When we expand this equation, it became \(\text{sin}^2 x + 2 \text{sin} x \text{cos} x + \text{cos}^2 x = k\). This allowed us to apply the Pythagorean identity and solve step-by-step by breaking the problem down into simpler parts. Expanding is crucial because it often reveals hidden relationships between terms, making it easier to apply further algebraic or trigonometric identities.