Two waves add up to one.

A cosine wave plus a sine wave of the same period is not a messy new shape. It is one wave, taller and shifted, and a right-angled triangle tells you exactly how.

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$3\cos\theta$ has height $3$ and peaks at $\theta = 0$. $4\sin\theta$ has height $4$ and peaks a quarter-turn later. What do you get if you add them?

At every $\theta$, stack one height on the other. The result is another wave with the same period, just taller and shifted along: its first peak comes between the two peaks.

Its height $R$ and its shift $\alpha$ come from the right-angled triangle with sides $3$ and $4$: $R = \sqrt{3^2 + 4^2} = 5$ and $\tan\alpha = \tfrac{4}{3}$, so $\alpha = 0.9272\ldots$

Now the wave is easy to read. Its maximum is $5$, at $\theta = \alpha = 0.927$. And $3\cos\theta + 4\sin\theta = 2$ becomes $\cos(\theta - \alpha) = 0.4$, so for $0 \le \theta < 2\pi$:

Match the coefficients, read the triangle.

The addition formulae themselves, double angles, the other R forms, modelling tides and sound, and twelve graded exam questions with full worked solutions are waiting.

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