Before the method: the idea behind it
The functions that come back around
The exponential was special because differentiating left it unchanged. Sine and cosine are special in a related but stranger way: differentiating turns each into the other, and after four steps you are back where you started. That closed loop is why these two functions describe everything that repeats, from tides and daylight to sound and alternating current.
Part one
Where the wave comes from
Put a point on a circle and spin it at a steady rate. Now watch only its height.
The height rises, slows near the top, falls, slows near the bottom, and repeats forever. Plot that height against the angle turned and you get the sine curve. It isn’t a shape someone invented; it is what circular motion looks like from the side. That is the whole reason sine and cosine turn up whenever something rotates, orbits, oscillates or cycles.
Watch the wheel below. The blue dot’s height traces sine, and the red marker shows how fast that height is changing at that instant.
Notice where the tangent is flattest: exactly at the peaks and troughs, where the height has stopped changing. And where it is steepest: as the curve crosses zero.
Part two
The gradient of sine is cosine
Track the gradient of the sine curve at every point, and plot those gradients as a curve of their own.
At x = 0 the sine curve is rising as steeply as it ever does, and its gradient is 1. At the peak the gradient is 0. Going down through π the gradient is −1. At the trough it is 0 again. Plot those values and a familiar shape appears: it is the cosine curve, exactly.
Drag or press play. The lower curve is built from the gradients of the upper one.
The same experiment on cosine gives the negative of sine. Differentiating four times returns you to the start, which is the closed loop that makes these functions so well behaved.
One warning that costs marks every year: these formulas are only true when x is in radians. The derivative of sin x measured in degrees is not cos x, it is \(\frac{\pi}{180}\cos x\), because the input has been rescaled. All calculus with trigonometric functions is done in radians, and your ClassPad must be set to Rad mode before you start.
Reversing the cycle gives the integrals. The minus sign lands on the cosine, which is the single most common sign error in this topic. A quick check: differentiate −cos x and you get +sin x, so it is right.
Part three
Everything that repeats
A general model looks like \(y=A\sin(nt)+d\), and every letter has a physical meaning.
A is the amplitude, how far the quantity swings above and below its middle. The n controls how fast it cycles: the period is \(\frac{2\pi}{n}\), so a larger n means faster repetition. And d shifts the whole thing up to the average level. Fit those three to a tide, a temperature or a sound wave and calculus then tells you the rate of change at any moment.
Where this shows up in real life
Tides and daylight
Tide height and hours of daylight both cycle almost sinusoidally. Port authorities publish tide tables generated from models of exactly the form A sin(nt) + d, and differentiating tells them how fast the water is rising at any hour, which matters for when a ship can safely enter.
Sound, light and electricity
A pure musical note is a sine wave in air pressure. Mains electricity is a sine wave in voltage, cycling 50 times a second in Australia. Engineers work with the derivatives of these constantly, since a changing current is what induces a voltage in the first place.
Before you start
What to reach for
| The question involves… | Use |
|---|---|
| sin or cos of a multiple of x, such as sin 3x | Chain rule: the inner derivative comes out the front |
| x sin x, e^(−x) sin x, or similar products | Product rule, with the trig derivative as one factor |
| tan x | Write as sin x / cos x and use the quotient rule |
| integrating sin or cos of ax + b | Reverse the cycle and divide by a |
| amplitude, period, maximum, rate of change | Read A, n and d from the model, then differentiate |
Next: the working method, twenty practice questions, ten harder ones, and the ClassPad guide, starting with how to set radian mode.