Problem 36 - Calculus Single Variable

Question

Find constants \(A\) and \(B\) such that $$\sin (3 \theta)=A \sin (\theta)-B \sin ^{3}(\theta)$$ for every \(\theta\).

Step-by-Step Solution

Answer

A = 3, B = 4.

Step 1: Expand \(\sin(3\theta)\)

The triple angle formula for sine states that \(\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\). We will use this identity to compare it directly with the given expression \(A \sin(\theta) - B \sin^3(\theta)\).

Step 2: Equate Coefficients

From the identity \(\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\), it is clear that we can equate this to \(A \sin(\theta) - B \sin^3(\theta)\). Thus, we equate coefficients: \(A = 3\) and \(B = 4\).

Step 3: Verify the Solution

Substitute back the values of \(A\) and \(B\) into the original expression to verify: \(3\sin(\theta) - 4\sin^3(\theta)\). This matches the original expansion for \(\sin(3\theta)\), so the solution is verified.

Key Concepts

Understanding the Triple Angle Formula

The triple angle formula is a key trigonometric identity that allows us to express trigonometric functions of angles that are three times larger in terms of simpler functions. Specifically, for the sine function, the triple angle formula is expressed as: - \( \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \). This formula is derived using trigonometric identities and the addition formula for sine. By understanding this formula, you can simplify more complex trigonometric expressions. It is important to know this formula when dealing with various angles, as it allows trigonometric expressions to be reorganized into more manageable parts. This is especially helpful when comparing or equating expressions, such as in the exercise above. The formula essentially breaks down the sine of thrice the angle into linear and cubic components of sine, aiding in algebraic manipulations. Knowing this identity can deepen understanding of the relationships between angles and their trigonometric functions.

Exploring the Sine Function

The sine function is one of the fundamental trigonometric functions, which can be defined for any real number. In the context of a unit circle, the sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle. - The sine function has a range from -1 to 1 and is periodic, repeating every \(2\pi\) radians, or 360 degrees.- When graphed, it has a characteristic wave-like pattern.- It crosses the x-axis at \(0, \pi, 2\pi, \ldots\) and peaks at odd multiples of \(\pi/2\).Sine's periodic properties are deeply connected to circular motion and oscillations, which is why the understanding of this function is crucial in physics and engineering. In problems involving trigonometric identities, such as our exercise, recognizing these features helps in analyzing and manipulating expressions with the sine function. By expanding functions like \(\sin(3\theta)\), we can observe how the sine function's interplay with other harmonics leads to different trigonometric identities used for problem-solving.

The Method of Equating Coefficients

Equating coefficients is a powerful algebraic method used to find unknowns in expressions. This technique is particularly useful in dealing with polynomial equations or when comparing trigonometric identities. - In our exercise, we have \( \sin(3\theta) = A\sin(\theta) - B\sin^3(\theta) \). - Since the expression on both sides include \(\sin(\theta)\) and \(\sin^3(\theta)\), we separately equate the coefficients of identical terms. - We must ensure each corresponding coefficient is equal by the principle of identity.This means setting the coefficients of \(\sin(\theta)\) equal (\(A = 3\)) and the coefficients of \(\sin^3(\theta)\) equal (\(B = 4\)). This technique ensures the equivalence of two algebraic expressions and is a straightforward method to solve for unknown parameters. Equating coefficients provides not only an algebraic solution but also a clearer insight into the structure and symmetry of expressions, fostering a deeper understanding of algebraic relationships.