2z = 60^\circ + 360^\circ k \quad \textو \quad 2z = 120^\circ + 360^\circ k \quad \textلـ k \text عدد صحيح

Understanding & Solving the Congruent Angles Equation: 2z = 60° + 360°k and 2z = 120° + 360°k (k ∈ ℤ)

Introduction

Trigonometric equations involving angle congruences are fundamental in mathematics, especially in geometry, physics, and engineering. Equations like 2z = 60° + 360°k and 2z = 120° + 360°k (where k is any integer) describe infinite families of angles that satisfy specific angular relationships. In this SEO-optimized article, we explore these equations step-by-step, explain their significance, and offer practical insights into solving and applying them.

Understanding the Context

What Are the Equations?

We consider two primary trigonometric congruence equations:

1. 2z = 60° + 360°k
2. 2z = 120° + 360°k (k ∈ ℤ, i.e., integer values of k)

$2z = 60^\circ + 360^\circ k \quad \text{و} \quad 2z = 120^\circ + 360^\circ k \quad \text{لـ } k \text{ عدد صحيح}$

Key Insights

Here, z is a real variable representing an angle in degrees, and k is any integer that generates periodic, recurring solutions across the angular spectrum.

Step 1: Simplify the Equations

Divide both sides of each equation by 2 to isolate z:

1. z = 30° + 180°k
2. z = 60° + 180°k

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Final Thoughts

These simplified forms reveal a key insight: since 360° / 2 = 180°, all solutions of the original equations occur at intervals of 180°—the period of the sine and cosine functions divided by 2.

Step 2: Interpret the Solutions

For z = 30° + 180°k

This means:

When k = 0, z = 30°
When k = 1, z = 210°
When k = -1, z = -150°
And so on…

All solutions are spaced every 180°, all congruent modulo 360° to 30°.

For z = 60° + 180°k