Question

What value of \(k\) makes the equation \(\sin 5 x \cos x+\cos 5 x \sin x=2 \sin k x \cos k x\) true?

Short answer

The value of k is 3.

Step-by-step solution

Recognize Trigonometric Identities

Observe that the left-hand side can be simplified using the angle addition formula for sine. Recall that \(\bullet\) \(\bullet\) \(\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)\).This implies \(\bullet\) \(\sin(5x + x) = \sin(5x)\cos(x) + \cos(5x)\sin(x)\).

Simplify the Left-Hand Side

Use the angle addition formula to simplify \(\bullet\) \(\sin(5x + x) = \sin(6x) \). Thus, the equation \(\bullet\) \(\sin 5 x \cos x + \cos 5 x \sin x \) on the left side simplifies to \(\bullet\) \(\sin(6x) \).

Simplify the Right-Hand Side

Recognize that \(\bullet\) \(2 \sin k x \cos k x\) can be rewritten using the double angle identity for sine: \(\bullet\) \( \sin(2kx)\). Hence, the right-hand side becomes \(\bullet\) \(\sin(2kx) \).

Equate the Simplified Forms

Equate the simplified left-hand side and right-hand side: \(\bullet\) \(\sin(6x) = \sin(2kx)\).

Solve for k

To resolve \(\bullet\) \(\sin(6x) = \sin(2kx)\), the arguments of the sine functions must be equal for all x. This implies \(\bullet\) \(6x = 2kx\). Solving for \(\bullet\) \(k \) gives: \(\bullet\) \(6 = 2k\) which simplifies to \(\bullet\) \(k = 3\).

Key concepts

Angle Addition Formula

The angle addition formula is a key identity in trigonometry. It allows the simplification of expressions involving the sine and cosine of sums of angles. For instance, \[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \] This formula helps us break down more complex trigonometric expressions into more manageable parts.
Using this identity, we can tackle a variety of trigonometric equations.

Double Angle Identity

Another crucial trigonometric identity is the double angle identity. This identity helps simplify expressions involving twice an angle. For the sine function, the double angle identity is given by: \[ \sin(2a) = 2\sin(a)\cos(a) \]
This identity allows us to express the sine of double an angle in terms of the product of sine and cosine of the original angle. It can be particularly useful when solving equations or in various integration problems.

Solving Equations with Sine Functions

Solving trigonometric equations often involves using identities to simplify expressions. When we have something like \[ \sin(A)= \sin(B) \] it implies that \[ A = B + 2k\pi \] or \[ A = \pi - B + 2k\pi \] where k is any integer.
This provides multiple solutions based on the periodic nature of the sine function.
For trigonometric equations, finding such solutions requires recognizing which identities to use and applying them correctly. This step-by-step approach helps in isolating the variable and finding all possible solutions accurately.