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Chapter 6: Problem 70

In Exercises \(69-82,\) prove the given identities. $$\sin (x+x)=2 \sin x \cos x$$

Short Answer

Expert verified
The identity \(\sin (x+x)=2 \sin x \cos x\) can be proven by applying the sum formula for sine and combining like terms.

Step by step solution

01

Identify the identity

The given identity is \(\sin (x+x)=2 \sin x \cos x\), and the goal is to show that the left-hand side (LHS) is identical to the right-hand side (RHS).
02

Apply the sum formula

The formula for the sine of a sum, \(\sin(a + b) = \sin a \cos b + \cos a \sin b\), is applied to the LHS of the equation. When replacing \(a\) and \(b\) with \(x\), we get \(\sin(x + x) = \sin x \cos x + \cos x \sin x\).
03

Simplify the equation

Now, simplify the LHS of the equation by combining like terms: \(\sin x \cos x + \cos x \sin x = 2 \sin x \cos x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine of a Sum
Understanding the 'sine of a sum' is a key step in proving various trigonometric identities and solving complex trigonometric equations. It refers to a formula used when you are dealing with the sine function of the sum of two angles. The formula is stated as:
\[\begin{equation}\sin(a + b) = \sin a \cos b + \cos a \sin b\end{equation}\]
Applied to our exercise, where the sum is \(x+x\) or \(2x\), the formula simplifies to \(\sin(x + x) = \sin x \cos x + \cos x \sin x\). This identity proves incredibly useful when we encounter trigonometric expressions that can appear complex at first glance but are actually just an application of this foundational principle.
Simplifying Equations
Simplifying equations is all about reducing expressions to their most basic form without changing their value. It's an essential skill in mathematics, allowing us to understand and solve equations more effectively. In trigonometry, simplification often involves combining like terms, factoring, and using fundamental identities. In our case, the expression \(\sin x \cos x + \cos x \sin x\) can be simplified because the terms \(\sin x \cos x\) and \(\cos x \sin x\) are identical. Hence, they can be combined to \(2 \sin x \cos x\), which matches the right-hand side of our original equation, thereby completing the simplification process.
Proving Identities
Proving trigonometric identities is a critical aspect of advanced mathematics, involving verifying that two sides of an equation are identical for all values of the variables involved. To show that an identity is true, we often start by working with one side of the equation and transforming it step by step until it matches the other side. The goal is to use known identities, algebraic manipulation, and logical reasoning to arrive at the proof. In the given exercise, we prove that the identity \(\sin (x+x) = 2 \sin x \cos x\) holds by applying the sum formula to the left-hand side and then simplifying the equation to match the right-hand side.
Trigonometric Functions
Trigonometric functions are functions of an angle and are foundational to the study of trigonometry. The basic trigonometric functions include sine (sin), cosine (cos), and tangent (tan), among others. These functions are vital in describing the relationships between the angles and sides of triangles, especially right-angled triangles, and they have applications in various scientific fields, including physics, engineering, and astronomy. When working with these functions, familiarity with their properties and the various identities involving them, such as \(\sin^2 x + \cos^2 x = 1\), is important for solving exercises and proving identities like the one we discussed.

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Most popular questions from this chapter

Use a graphing utility to find the solutions of the given equations, in radians, that lie in the interval \([0,2 \pi)\). $$\sin 2 x=\cos 2 x$$

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Using \(a=b=x,\) find a formula for \(\cos 2 x\).

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