IB Mathematics Analysis and Approaches Topic SL…

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Why Circular Functions: Modelling Things That Repeat

Tides, hours of daylight, sound waves and a seat on a Ferris wheel all rise and fall in a pattern that repeats, and sine and cosine are the natural tools to describe that repetition.
These functions come straight from the unit circle: as a point goes round and round, its coordinates rise and fall forever, which is exactly the periodic behaviour we want to model.
The goal of this section is to read and build the graph of $a\sin(b(x+c)) + d$ and to connect each number in it to a feature you can see.

The Graphs Of Sin, Cos And Tan

As the angle sweeps round the unit circle, the $y$-coordinate traces $\sin x$ and the $x$-coordinate traces $\cos x$, each a smooth wave between $-1$ and $1$.
Both $\sin x$ and $\cos x$ have amplitude $1$ and repeat every $2\pi$, and $\cos x$ is just $\sin x$ shifted left by $\dfrac{\pi}{2}$.
The graph of $\tan x = \dfrac{\sin x}{\cos x}$ is different: it repeats every $\pi$ and shoots off to infinity at vertical asymptotes wherever $\cos x = 0$.

The General Form a sin(b(x + c)) + d

Every parameter in $a\sin(b(x + c)) + d$ controls one visible feature of the wave.
1. $a$ sets the amplitude $|a|$, which is half the distance from the maximum down to the minimum (a vertical stretch).
2. $b$ sets the period, which is $\dfrac{2\pi}{b}$; a larger $b$ squeezes the wave into a shorter repeat (a horizontal stretch of factor $\frac{1}{b}$).
3. $c$ gives the horizontal shift, moving the wave left by $c$.
4. $d$ gives the vertical shift, raising the centre line (the principal axis) to $y = d$.
From these, the maximum is $d + |a|$ and the minimum is $d - |a|$.

Example question

A curve has equation $y = a\sin(bx) + d$ with maximum value $7$ and minimum value $-1$, and period $\pi$. Find $a$, $d$ and $b$.

Solution

Amplitude is half the range: $a = \frac{7 - (-1)}{2} = 4$.
The midline is the average of max and min: $d = \frac{7 + (-1)}{2} = 3$.
The period is $\frac{2\pi}{b} = \pi$, so $b = 2$.

Transformations Step By Step

Reading $a\sin(b(x + c)) + d$ as a sequence of transformations of $y = \sin x$ keeps the changes clear.
Take $y = \sin x \to y = 3\sin 2x$ as an example: the $3$ is a vertical stretch of factor $3$, and the $2$ is a horizontal stretch of factor $\frac{1}{2}$.
The stretches change amplitude and period, while $c$ and $d$ then slide the whole curve sideways and up or down.

Example question

Describe the transformations that map $y = \cos x$ onto $y = 2\cos(x - \tfrac{\pi}{4}) + 1$.

Solution

The factor $2$ is a vertical stretch of factor $2$, so the amplitude becomes $2$.
The $-\frac{\pi}{4}$ inside is a horizontal shift to the right by $\frac{\pi}{4}$.
The $+1$ is a vertical shift up by $1$, raising the midline to $y = 1$.

Real-Life Contexts

To model a repeating real situation, read three things off the description: the midline is the average level, the amplitude is half the full range, and the period is the time to repeat once.
Choose sine or cosine to match the starting point, then find $b$ from the period using $b = \dfrac{2\pi}{\text{period}}$.
Be aware that a calculator's regression tool may not return the exact form $a\sin(b(x + c)) + d$, so you sometimes rearrange its output.

Example question

A Ferris wheel has diameter $40\,\mathrm{m}$, its lowest point is $2\,\mathrm{m}$ above the ground, and it completes one turn every $120\,\mathrm{s}$. Write a model $h(t)$ for the height of a capsule that starts at the lowest point.

Solution

The midline is the centre height: $d = 2 + 20 = 22$, and the amplitude is the radius: $a = 20$.
The period is $120\,\mathrm{s}$, so $b = \frac{2\pi}{120} = \frac{\pi}{60}$.
Starting at the lowest point means starting at the minimum, so use a negative cosine: $$h(t) = 22 - 20\cos\left(\frac{\pi}{60}t\right)$$

Common Mistake

Writing the period as $2\pi b$ instead of $\frac{2\pi}{b}$.
Taking the amplitude as the full range rather than half of it.
Giving a negative amplitude; amplitude is $|a|$, so it is never negative even when $a$ is.
Shifting the wrong way for $c$, or forgetting to factor out $b$ so that $c$ is read correctly.
Working in degrees when the graph and question are in radians.

Active recall

What are the amplitude and period of $y = \sin x$, and how does $\tan x$ differ?
In $y = a\sin(b(x + c)) + d$, what feature does each of $a$, $b$, $c$ and $d$ control?
How do you find the period from the value of $b$?
Describe the transformations mapping $y = \sin x$ onto $y = 3\sin 2x$.
How do you read the amplitude, midline and period from a real-world repeating situation?

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Notes

Why Circular Functions: Modelling Things That Repeat

The Graphs Of Sin, Cos And Tan

The General Form a sin(b(x + c)) + d

Transformations Step By Step

Real-Life Contexts

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Notes

Why Circular Functions: Modelling Things That Repeat

The Graphs Of Sin, Cos And Tan

The General Form a sin(b(x + c)) + d

Transformations Step By Step

Real-Life Contexts

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Notes

Why Circular Functions: Modelling Things That Repeat

The Graphs Of Sin, Cos And Tan

The General Form a sin(b(x + c)) + d

Transformations Step By Step

Real-Life Contexts

IB Mathematics Analysis and Approaches SL 3.7—circular Functions, Graphs, Composites, Transformations Notes

Why Circular Functions: Modelling Things That Repeat

The Graphs Of Sin, Cos And Tan

The General Form a sin(b(x + c)) + d

Transformations Step By Step

Real-Life Contexts