{"id":61497,"date":"2026-03-14T04:43:02","date_gmt":"2026-03-13T19:43:02","guid":{"rendered":"https:\/\/phys-edu.net\/wp\/?p=61497"},"modified":"2026-03-14T04:43:02","modified_gmt":"2026-03-13T19:43:02","slug":"make-sense-of-tricky-ac-circuits-visualizing-the-addition-of-trigonometric-functions-with-animation-scratch","status":"publish","type":"post","link":"https:\/\/phys-edu.net\/wp\/?p=61497&lang=en","title":{"rendered":"Make Sense of Tricky AC Circuits! Visualizing the Addition of Trigonometric Functions with Animation (Scratch)"},"content":{"rendered":"

I’m Ken Kuwako, your Science Trainer. Every day is an experiment!<\/strong><\/span><\/p>\n

Electricity powers our modern lives, but did you know that the “alternating current” (AC) humming through your home is actually a beautiful symphony of overlapping waves? The key to unlocking the complex maze of RLC circuits found in smartphones and appliances lies hidden within a mathematical gem: the harmonic addition theorem (or the method of combining trigonometric functions). However, just staring at formulas in a textbook rarely brings the concept to life. Today, I want to share the excitement of seeing math and physics click together through an animation I created to make these invisible electrical movements visible!<\/p>\n

The Powerhouse of the Electric World: The Harmonic Addition Theorem<\/h3>\n

When diving into AC theory and RLC circuits\u2014those systems that combine resistors, inductors, and capacitors\u2014you inevitably run into the need to combine sine and cosine waves. Without understanding how this formula is derived or what it means geometrically, many students find themselves asking, “Wait, why are we doing this calculation again?” and losing their way.<\/p>\n

Let\u2019s start with the fundamental formula. I\u2019ve written it using \u03c9t (omega-t) for the angle, as this is the standard form we use in physics class.<\/p>\n

The Harmonic Addition Theorem:<\/p>\n

\"\"<\/p>\n

The Secret Identity of the Formula: It’s All About Rotation<\/h3>\n

What is this equation actually trying to tell us? Let\u2019s look at it through a geometric lens. When you compare it to a standard sine wave (sin\u03c9t), the magic really starts to show.<\/p>\n

\"\" The Geometric Meaning<\/p>\n

\"\"<\/p>\n

Looking at this diagram, you can see that the “combined wave” is essentially the shadow of a vector spinning in a circle with radius r. Imagine a sine wave that starts its rotation with a slight head start (an initial phase shift of \u03b1). Since static images in textbooks make this “movement” hard to grasp, I\u2019ve built an animation for you! It should give you an intuitive feel for how the math physically builds the wave shape.<\/p>\n

Give it a try here:<\/p>\n

https:\/\/scratch.mit.edu\/projects\/303808808\/<\/a><\/p>\n