Shifts, stretches and squeezes of sine and cosine

Trigonometry, grade 10 (5 units) Lesson 7 of 21Learning

Shifts, stretches and squeezes of sine and cosine

A general sine and cosine

Each number in the function $\displaystyle {y}={A\sin(ax+b)+B}$ changes the graph of $\displaystyle {y}={\sin x}$ in a different way:

$\displaystyle B$ – a vertical shift: $\displaystyle B$ units up ( $\displaystyle B<0$ – down). The graph swings about the line $\displaystyle {y}={B}$ .
$\displaystyle A$ – a vertical stretch: the height above $\displaystyle {y}={B}$ and below it grows $\displaystyle |A|$ times. The number $\displaystyle |A|$ is the amplitude. When $\displaystyle A<0$ the graph is also reflected.
$\displaystyle a$ – a horizontal squeeze by $\displaystyle |a|$ : a full turn of $\displaystyle ax$ ends already after $\displaystyle \frac{2\pi}{|a|}$ . This is the period:

\displaystyle T

\displaystyle =

\displaystyle \frac{2\pi}{|a|}

$\displaystyle b$ – a horizontal shift: $\displaystyle {ax+b}={a\left(x+\frac{b}{a}\right)}$ , so the graph moves $\displaystyle \frac{b}{a}$ to the left ( $\displaystyle \frac{b}{a}<0$ – to the right).

The range: $\displaystyle B-|A|\leq y\leq B+|A|$ . The very same rules hold for $\displaystyle {y}={A\cos(ax+b)+B}$ .

Example: reading the formula

The example below reads the properties of $\displaystyle {g(x)}={2\sin\left(2x-\frac{\pi}{2}\right)+1}$ from the formula. On the board: $\displaystyle g$ , and dashed – $\displaystyle \sin x$ .

$\displaystyle {B}={1}$ : the graph swings about $\displaystyle {y}={1}$ .
$\displaystyle {A}={2}$ : the amplitude is 2, so the range is $\displaystyle -1\leq y\leq3$ .
$\displaystyle {a}={2}$ : the period is $\displaystyle {\frac{2\pi}{2}}={\pi}$ .
$\displaystyle {2x-\frac{\pi}{2}}={2\left(x-\frac{\pi}{4}\right)}$ : a shift of $\displaystyle \frac{\pi}{4}$ to the right.

\displaystyle f(x)=\sin x

\displaystyle g(x)=2\sin\left(2x-\frac{\pi}{2}\right)+1

Finding the function from its graph

On the board: the graph of a function of the form $\displaystyle A\sin(ax+b)+B$ , with a maximum point $\displaystyle M$ and the minimum point $\displaystyle N$ after it. The example below finds its formula from these points.

$\displaystyle B$ – the middle of the range: $\displaystyle {\frac{3+(-1)}{2}}={1}$ .
The amplitude – half the height of the range: $\displaystyle {\frac{3-(-1)}{2}}={2}$ .
From the maximum to the minimum is half a period: $\displaystyle {\frac{3\pi}{4}-\frac{\pi}{4}}={\frac{\pi}{2}}$ , so the period is $\displaystyle \pi$ and $\displaystyle {a}={\frac{2\pi}{\pi}}={2}$ .

\displaystyle f(x)=2\sin2x+1

\displaystyle M=(\frac{\pi}{4},3)

\displaystyle N=(\frac{3\pi}{4},-1)

The shift is left to find. It has more than one answer:

as a sine: $\displaystyle \sin$ reaches its maximum when its argument is $\displaystyle \frac{\pi}{2}$ . At $\displaystyle {x}={\frac{\pi}{4}}$ : $\displaystyle {2\cdot\frac{\pi}{4}}={\frac{\pi}{2}}$ – no shift is needed, $\displaystyle {y}={2\sin2x+1}$ .
as a cosine: $\displaystyle \cos$ reaches its maximum when its argument is $\displaystyle 0$ , so the graph is $\displaystyle \cos2x$ shifted $\displaystyle \frac{\pi}{4}$ to the right: $\displaystyle {y}={2\cos\left(2x-\frac{\pi}{2}\right)+1}$ .

Both formulas describe the same graph, because $\displaystyle {\cos\left(2x-\frac{\pi}{2}\right)}={\cos\left(\frac{\pi}{2}-2x\right)}={\sin2x}$ .

Exercise 1

The function is $\displaystyle {f(x)}={3\sin2x-1}$ .

Mark on the board its maximum and minimum points in the part shown.

\displaystyle f(x)=3\sin2x-1

Write the amplitude, the period and the range of $\displaystyle f$ , and how they show in the points you marked.

Exercise 2

On each board – the graph of one transformation of $\displaystyle \sin x$ or of $\displaystyle \cos x$ .
Write its formula on the board.

\displaystyle y=\,?

\displaystyle y=\,?

\displaystyle y=\,?

\displaystyle y=\,?

Exercise 3

Write two formulas for the graph on the board: one as a transformation of $\displaystyle \sin x$ , and one as a transformation of $\displaystyle \cos x$ .

\displaystyle y=\,?

Exercise 4

The period of $\displaystyle {y}={\sin ax}$ ( $\displaystyle a>0$ ) is $\displaystyle \frac{\pi}{3}$ .
Solve the equation to find $\displaystyle a$ .

Formulas: trigonometry, grade 10 (5 units)

Open as a page

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 $\displaystyle b$ )
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$