From Ratio to Function

For most of your mathematical life $\sin$ has been a ratio — a number you get out of a triangle. Turning it into a function means letting the angle be the input and watching what comes out as the angle keeps growing past the triangle’s limits, and the graph you get is the reason trigonometry describes tides, sound, daylight, and alternating current.

The unit circle does the work

Put a point on a circle of radius 1 and let it travel counterclockwise. At angle $x$ the point is at $(\cos x,\sin x)$. So:

• $\sin x$ is the height of the point above the horizontal axis;
• $\cos x$ is its horizontal position.

Now let $x$ keep growing past $90^{\circ }$, past $180^{\circ }$, past $360^{\circ }$. The triangle stopped making sense a while ago; the circle does not care. Plot the height against the angle and a wave appears, repeating every $360^{\circ }$ because the point has come back to where it started.

That repetition is what periodic means, and it is the property that makes these functions useful for anything that cycles.

Reading a periodic graph

Feature	What it is	How to find it
Period	The horizontal length of one full cycle	Peak to the next peak
Amplitude	Half the distance from maximum to minimum	$\frac{\max -\min }{2}$
Axis	The horizontal centre line	$\frac{\max +\min }{2}$
Phase shift	How far the cycle starts from the standard position	Compare a peak with where the parent’s peak sits

A function is sinusoidal when it has that shape — a smooth, symmetric wave. Many periodic functions are not: a sawtooth voltage and the number of daylight minutes rounded to whole days both repeat without being sinusoidal. Periodic is the general family; sinusoidal is the smooth member of it.

The four parameters again

$y=a\sin (k(x-d))+c$

The same $a$, $k$, $d$, $c$ as everywhere else in this course, with trigonometric names attached:

• $|a|$ is the amplitude — how tall the wave is.
• $k$ compresses horizontally, so the period becomes $\frac{360^{\circ }}{|k|}$. This is the one that surprises people: a larger $k$ makes a shorter period.
• $d$ is the phase shift, moving the wave sideways.
• $c$ raises the axis to $y=c$.

Predict each before you graph it in Using Desmos, the same routine as Transformations of Functions. The habit is the point: the parameters do not change meaning when the parent function changes, and noticing that is most of what Grade 11 functions is for.

Going from a situation to an equation

Given real measurements — a tide table, a Ferris wheel, hours of daylight — the route is always the same:

1. Find the maximum and minimum. Those give the amplitude and the axis.
2. Find how long one cycle takes. That gives $k$.
3. Decide where the cycle starts. That gives $d$, and choosing $\sin$ or $\cos$ can make $d$ zero — take the easier one.
4. Check with a data point you did not use to build it.

Step 4 is the one students skip and the one that catches a wrong period. The Ferris Wheel and The Tide Problem are where you do all four under real conditions.

Curriculum connection

D2.3

make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from $0^{\circ }$ to $360^{\circ }$ and the corresponding sine ratios or cosine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function $f(x)=\sin x$ or $f(x)=\cos x$, and explaining why the relationship is a function

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D2.5

determine, through investigation using technology, the roles of the parameters $a$, $k$, $d$, and $c$ in functions of the form $y=af(k(x-d))+c$, where $f(x)=\sin x$ or $f(x)=\cos x$ with angles expressed in degrees, and describe these roles in terms of transformations on the graphs of $f(x)=\sin x$ and $f(x)=\cos x$ (i.e., translations; reflections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes) Sample problem: Investigate the graph $f(x)=2\sin (x-d)+10$ for various values of $d$, using technology, and describe the effects of changing $d$ in terms of a transformation.

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D3.2

identify periodic and sinusoidal functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range Sample problem: Using data from Statistics Canada, investigate to determine if there was a period of time over which changes in the population of Canadians aged 20–24 could be modelled using a sinusoidal function.

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