Adding Sine and Cosine

Always another sine wave

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $a\sin x + b\cos x = R\sin(x + \alpha)$, with $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = b/a$.
• So its maximum is $\sqrt{a^2 + b^2}$.
• Same-frequency waves always combine into one wave of that frequency.

Steps

1. A surprising sum
2. Spinning arrows
3. Maximum value
4. Find the shift
5. Minimum value
6. Period of a combination
7. When is the peak?