Angle Addition Discovery

Trigonometry · Axiom Academy

INTRO Angle Addition Discovery Does sin(A + B) = sin A + sin B? Let's investigate this tempting assumption and discover why trigonometry needs special formulas. When we first learn about functions, we might think that adding angles works like this: But is this actually true ? Let's put it to the test! Use the sliders to try different angle combinations. Watch what happens to the actual value versus what the formula would give us. Let's look at a specific example that clearly shows the formula doesn't work. Notice that sin(90°) = 1 , but our wrong formula gives us a value greater than 1 ! That's impossible for the sine function, which must always be between -1 and 1. The unit circle helps us visualize why angles don't add the simple way. The sine values are heights , and adding heights doesn't equal the height of the sum angle. Why We Need Angle Addition Formulas The intuitive guess sin(A + B) = sin A + sin B is false . We proved it with counterexamples, saw the numerical differences, and understood geometrically why it fails. Sine doesn't distribute over addition. The correct formula involves both sine AND cosine: This formula captures the true geometric relationship between angles on the unit circle. Angle addition formulas are fundamental to trigonometry. They let us find exact values for angles like 75° (45° + 30°), simplify complex expressions, and solve trigonometric equations. They're essential tools throughout calculus and physics!

This is the written version of the interactive lesson above. See the full Trigonometry course.