R \sin(2z + \alpha) = \sin(2z) + \sqrt3 \cos(2z)

Understanding R·sin(2z + α) = sin(2z) + √3·cos(2z): A Complete Guide

When tackling trigonometric equations like R·sin(2z + α) = sin(2z) + √3·cos(2z), recognizing the power of trigonometric identities can simplify complex expressions and unlock elegant solutions. This article explains how to rewrite the right-hand side using the sine addition formula, identifies the amplitude R and phase shift α, and explores practical applications of this identity in mathematics and engineering.

Understanding the Context

The Goal: Rewriting sin(2z) + √3·cos(2z) in the form R·sin(2z + α)

To simplify the expression
sin(2z) + √3·cos(2z)
we use the standard identity:
R·sin(2z + α) = R·(sin(2z)cos(α) + cos(2z)sin(α))

This expands to:
R·cos(α)·sin(2z) + R·sin(α)·cos(2z)

By comparing coefficients with sin(2z) + √3·cos(2z), we get:

R·cos(α) = 1
R·sin(α) = √3

$R \sin(2z + \alpha) = \sin(2z) + \sqrt{3} \cos(2z)$

Key Insights

Step 1: Calculate the Amplitude R

Using the Pythagorean identity:
R² = (R·cos(α))² + (R·sin(α))² = 1² + (√3)² = 1 + 3 = 4
Thus,
R = √4 = 2

This amplitude represents the maximum value of the original expression.

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Final Thoughts

Step 2: Determine the Phase Shift α

From the equations:

cos(α) = 1/R = 1/2
sin(α) = √3/R = √3/2

These values correspond to the well-known angle α = π/3 (or 60°) in the first quadrant, where both sine and cosine are positive.

Final Identity

Putting it all together:
sin(2z) + √3·cos(2z) = 2·sin(2z + π/3)

This transformation converts a linear combination of sine and cosine into a single sine wave with phase shift — a fundamental technique in signal processing, wave analysis, and differential equations.

Why This Identity Matters

Simplifies solving trigonometric equations: Converting to a single sine term allows easier zero-finding and period analysis.
Enhances signal modeling: Useful in physics and engineering for analyzing oscillatory systems such as AC circuits and mechanical vibrations.
Supports Fourier analysis: Expressing functions as amplitude-phase forms underpins many Fourier series and transforms.
Improves computational efficiency: Reduces complexity when implementing algorithms involving trigonometric calculations.