Periodic processes

Trigonometry, grade 10 (5 units) Lesson 8 of 21Learning

Periodic processes

A periodic process

A process that repeats itself every fixed length of time is a periodic process, and that length of time is its period:

a car on a Ferris wheel – its height repeats every turn;
the water level in a harbour – high and low tide, twice a day;
the length of the day – a period of a year.

When the process rises and falls smoothly, it is described by a function $\displaystyle {y}={A\sin(ax)+B}$ or $\displaystyle {y}={A\cos(ax)+B}$ :

$\displaystyle B$ – the average level, halfway between the maximum and the minimum;
$\displaystyle |A|$ – how far it rises above the average and falls below it;
$\displaystyle {a}={\frac{2\pi}{T}}$ , where $\displaystyle T$ is the period.

There are periodic processes that are not a sine as well: a runner going round a track, for example, comes back to where she started every lap, but her distance from the start does not rise and fall like a sine wave.

Example: a Ferris wheel

A Ferris wheel has radius $\displaystyle 20$ m, and its centre is $\displaystyle 22$ m above the ground.
The wheel makes a full turn in $\displaystyle 12$ minutes.
At time $\displaystyle {t}={0}$ our car is at the lowest point.

The example below finds the car’s height as a function of time, and when it is $\displaystyle 32$ m high.

The average height is $\displaystyle {B}={22}$ , and the car rises and falls $\displaystyle 20$ m about it.
At $\displaystyle {t}={0}$ the car is at the bottom. $\displaystyle -\cos$ starts at its minimum, so $\displaystyle {h(t)}={22-20\cos(at)}$ .
The period is $\displaystyle 12$ , so $\displaystyle {a}={\frac{2\pi}{12}}={\frac{\pi}{6}}$ :

\displaystyle h(t)=22-20\cos\left(\frac{\pi}{6}t\right)

\displaystyle h(x)=22-20\cos\left(\frac{\pi}{6}x\right)

\displaystyle y=32

\displaystyle A=(4,32)

\displaystyle B=(8,32)

On the board: $\displaystyle h$ over two turns ( $\displaystyle x$ – the time in minutes) and the line $\displaystyle {y}={32}$ . In the first turn:

$\displaystyle {22-20\cos\left(\frac{\pi}{6}t\right)}={32}$ , that is $\displaystyle {\cos\left(\frac{\pi}{6}t\right)}={-\frac{1}{2}}$ ;
between $\displaystyle 0$ and $\displaystyle 2\pi$ : $\displaystyle {\frac{\pi}{6}t}={\frac{2\pi}{3}}$ or $\displaystyle {\frac{\pi}{6}t}={\frac{4\pi}{3}}$ ;
that is $\displaystyle {t}={4}$ or $\displaystyle {t}={8}$ : the car is $\displaystyle 32$ m high after $\displaystyle 4$ minutes on the way up, and after $\displaystyle 8$ minutes on the way down – the points $\displaystyle A$ and $\displaystyle B$ .

Exercise 1

The depth of the water in a harbour, in metres, $\displaystyle t$ hours after midnight, is $\displaystyle {d(t)}={3+2\sin\left(\frac{\pi}{6}t\right)}$ .
We look at one day: $\displaystyle 0\leq t\leq24$ .

Mark on the board the high tides (maximum) and the low tides (minimum) of the day ( $\displaystyle x$ – the hours).

\displaystyle d(x)=3+2\sin\left(\frac{\pi}{6}x\right)

When is the water exactly $\displaystyle 4$ m deep?

A ship enters the harbour only when the water is at least $\displaystyle 2$ m deep. When can it enter?

Exercise 2

Rangers in a nature reserve count the rabbits at the start of every month, from 1.1.2000 ( $\displaystyle {t}={0}$ ).
The number rises and falls periodically: the points on the board are some of the counts ( $\displaystyle x$ – the months from 1.1.2000).

Write on the board a function of the form $\displaystyle A\sin(ax)+B$ that passes through all the points.

\displaystyle (0,1000)

\displaystyle (3,1300)

\displaystyle (6,1000)

\displaystyle (9,700)

\displaystyle (12,1000)

\displaystyle (15,1300)

\displaystyle (18,1000)

\displaystyle (21,700)

\displaystyle (24,1000)

When in 2002 ( $\displaystyle 24\leq t\leq36$ ) will there be $\displaystyle 1{,}150$ rabbits in the reserve?

In which months will it happen?

Formulas: trigonometry, grade 10 (5 units)

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The unit circle (radians)

The unit circle
$\displaystyle x^2+y^2=1$
The point of the number $\displaystyle t$ on the unit circle
$\displaystyle P(t)=(\cos t,\ \sin t)$
All the numbers with the same point as $\displaystyle t$ ( $\displaystyle k$ an integer)
$\displaystyle t+2\pi k$
Radians and degrees
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle \ell$ $\displaystyle =$ $\displaystyle \alpha R$
The area of a sector of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha R^2$

Special values

$\displaystyle \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}$
$\displaystyle \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$
$\displaystyle \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$
$\displaystyle \tan\frac{\pi}{6}=\frac{\sqrt{3}}{3},\quad\tan\frac{\pi}{4}=1,\quad\tan\frac{\pi}{3}=\sqrt{3}$

Identities

The basic identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^2\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \sin(\alpha+2\pi k)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \tan(\alpha+\pi k)$ $\displaystyle =$ $\displaystyle \tan\alpha$

Graphs of trigonometric functions

A general sine (and a cosine likewise)
$\displaystyle y=A\sin(ax+b)+B$
The period of $\displaystyle \sin(ax+b)$ and of $\displaystyle \cos(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{2\pi}{|a|}$
The range of $\displaystyle A\sin(ax+b)+B$
$\displaystyle B-|A|\leq y\leq B+|A|$
The period of $\displaystyle \tan(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\pi}{|a|}$

Trigonometric equations ($k$ an integer)

The solutions of $\displaystyle {\sin x}={\sin\alpha}$
$\displaystyle x=\alpha+2\pi k,\quad x=\pi-\alpha+2\pi k$
The solutions of $\displaystyle {\cos x}={\cos\alpha}$
$\displaystyle x=\pm\alpha+2\pi k$
The solutions of $\displaystyle {\tan x}={\tan\alpha}$
$\displaystyle x=\alpha+\pi k$

Trigonometry in a triangle (degrees)

In a right-angled triangle: $\displaystyle a$ the leg opposite $\displaystyle \alpha$ , $\displaystyle b$ the leg beside it, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha=\frac{a}{c},\quad\cos\alpha=\frac{b}{c},\quad\tan\alpha=\frac{a}{b}$
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
The area of a triangle ( $\displaystyle \alpha$ – the angle between the sides $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}bc\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ and $\displaystyle \gamma$ – the angles at the side $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^2\sin\beta\sin\gamma}{2\sin(\beta+\gamma)}$
The law of sines ( $\displaystyle R$ – the radius of the circle that circumscribes the triangle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between the sides $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$