Graphs of Trigonometric Functions
$OP$ αှα့် positive $x$-axis αြားαှိαောα့် $(\theta)$ cosine ratio $(\cos \theta)$ αα် $OP$ αှα့် unit circle αြα်αွားαော
α‘αှα် $P(x,y)$ ၏ $x$-coordinate αα်αိုးαြα
်αြီး sine ratio $(\sin \theta)$ αα် $y$-coordinate αα်αိုးαြα
်αြောα်း
αိαှိαဲ့αြီး αြα
်αα်။
α‘αိုαါ $\sin \theta$ αှα့် $\cos \theta$ αα် $\theta$ αေါ်αူαα်၍αြောα်းαဲαေαα် αြα ်αα်αို unit circle αွα် α‘αွα်ααူ αိαှိαိုα်αα်။ $OP$ αα ်αα်αှα့်αα်αြα်း $ 0^{\circ} \le \theta \le 360^{\circ}$ αွα် $\sin \theta$ αှα့် $\cos \theta$ αα် $\theta$ αေါ်αူαα်၍αြောα်းαဲ αေαော်αα်း αα ်αα်αြα့်αြီးαောα် αောα်αွα် αူααα ်αα်α αα်αိုးαျားαိုαာ αြα်αα်αောα်αှိαာαα် αြα ်αα်။ αို့αို့ αα ်αα်αြα့်αိုα်း αα်αိုးαူαေαာαျားαို့ αြα်αα်αောα်αှိαာαော $\sin \theta$ αှα့် $\cos \theta$ αို့αို periodic function αျားαုαေါαြီး ααူαီαော αα်αိုးαျား ααှိα ေαα့် αα ်αα် $360^{\circ} \text{or } 2\pi$ αို $\sin \theta$ αှα့် $\cos \theta$ αို့၏ period αုαေါ်αα်။
unit circle ၏ α‘αα်းαိုα်းαို $(1,0)$ αေαာαှ αြα်αောα်၍ α‘αြောα့်α‘αိုα်းαားαိုα်αα်αု αူααα်။ α‘αα်းαိုα်းαေါ်αှိ $\theta$ ၏ αα်αိုးαှα့် αα်αိုα်αာ Trigonometric function value ၏ αα်αိုးαျားαို αှိုα်းαှα်αော်αြαြα်းαြα့် αα်αှα်αာαော Trigonometric Function ၏ graph αို ααှိαα်αြα ်αα်။ α‘ောα်αါ applet αွα် trigonometric function αα ်αုαျα်းα ီ၏ $0 \le \theta \le 2\pi$ α‘αွα်း ααှိαာαα့် trigonometric graph αα ်αုα ီαို αေ့αာαိုα်αါαα်။
The Graph of $\ { y=\sin\theta}$
terminal side αို aticlockwise direction αြα့် αှα့်αα့်α‘αါ $\sin \theta$ ၏ graph αα် positive $x$-axis
αα်αွα် αα
်αα်αြα့်αိုα်း wave αα
်αု αြα
်αာαြီး clockwise direction αြα့် αှα့်αα့်α‘αါ $\sin \theta$ ၏ graph
αα် negative $x$-axis αα်αွα် αα
်αα်αြα့်αိုα်း wave αα
်αု αြα
်αာαα်αြα
်αα်။ αို့αြောα့် $y=\sin\theta$ ၏
graph αို sine wave (αို့) sinusoidal wave αုαα်းαေါ်αα်။ sine wave αα
်αု αြα
်αα်αိုα‘α်αော $\theta$ αα်αိုးαို
period αုαေါ်αα်။ αို့αြောα့် $y=\sin\theta$ ၏ period αα် $360^{\circ} \text{or } 2\pi$ αြα
်αα်။
$y=\sin\theta$ ၏ maximum value αα် $1$ αြα ်αြီး minimum value αα် $-1$ αြα ်αα်။ αို့αြောα့် αα်αα့် $\theta$ αα်αိုးα‘αွα် ααို $-1\le \sin\theta\le 1$ αြα ်αα်။ α‘αိုαါαြားαိုα်းαို sine function ၏ range αု αေါ်αα်။ $y=\sin\theta$ αα် wave αွေ့αျားαာαျα်း (line of propagation) αှ α‘αြα့်αုံးα‘αှα်αိ $1$ unit αှိαြီး α‘αိα့်αုံးα‘αှα်αို့αα်း $1$ unit αှိαα်။ αို့αို့ line of propagation αှ α‘αြα့်αုံးα‘αှα် (αို့ααုα်) line of propagation αှ α‘αိα့်αုံးα‘αှα် αို့ α‘αွာα‘αေးαို amplitude αုαေါ်αα်။ α‘αြားαα ်αα်းαိုααော် α‘αြα့်αုံးα‘αှα် αှα့် α‘αိα့်αုံးα‘αှα် αှα ်αုαြား α‘αွာα‘αေး၏ αα ်αα်αို amplidude αု αေါ်αိုα်αα်။
The Graph of $\ { y=a\sin\theta}$
$y=\sin\theta$ αွα် $-1\le \sin\theta\le 1$ αြα
်αောαြောα့် $y=a\sin\theta$
αွα် $-a\le a\sin\theta\le a$ αြα
်αα်။ αို့αြောα့် ...
αုαှα်αားαိုα်αα်။
αို့αြောα့် ...
$y=\sin\theta\Rightarrow$ amplitude $=1$
$y=2\sin\theta\Rightarrow$ amplitude $=2$
$y=\displaystyle\frac{1}{2}\sin\theta\Rightarrow$ amplitude $=\displaystyle\frac{1}{2}$
Grade 11 αွα် αေ့αာαဲ့αြီးαြα ်αော Trasformation αှုαောα့်αှαြα့်αျှα် $y=a\sin\theta$ αα် $y=\sin\theta$ αို vertical scaling αြုαုα်αိုα်αြα်း αြα ်αြီး scale factor $=a$ αြα ်αα်αု αှα်αားαိုα်αα်။ αို့αြောα့် $y=\sin\theta$ αို vertical scaling αြုαုα်αျှα် scale factor αα် amplitude αြα ်αα်။
The Graph of $\ { y=\sin(b\cdot\theta})$
αα်αα်၍ α‘ောα်αါ diagram αို αေ့αာαြα့်αါ။
period of $y=\sin\theta\Rightarrow 2\pi$
$y=\sin 2\theta $ αα် $y=\sin \theta $ αို horizontal scaling αြုαုα်αိုα်αြα်း αြα
်αြီး scale factor αှာ
$\displaystyle\frac{1}{2}$ αြα
်αα်။ $y=\sin 2\theta $ ၏ period (the value of $\theta$ to form one complete cycle)
αှာ $\pi$ αြα
်αα်αို αွေ့ααα်။
α‘αားαူαα် $y=\sin \displaystyle\frac{1}{2}\theta $ αα် $y=\sin \theta $ αို horizontal scaling αြုαုα်αိုα်αြα်း αြα
်αြီး
scale factor αှာ $2$ αြα
်αα်။ $y=\sin \displaystyle\frac{1}{2}\theta $ ၏ period αှာ $4\pi$ αြα
်αα်။
αို့αြောα့် $y=\sin \theta $ αို horizontal scaling αြုαုα်αိုα်αြα်းαြα့် period αို α‘αြောα်းα‘αဲ αြα ်α ေαြီး αြောα်းαဲαွားαော period αα်αိုးαှာ scale factor $\times 2\pi\ $ αြα ်αα်αို αွေ့ααα်။ α‘ောα်αါα‘αိုα်း αျေαုαျ αှα်αားαိုα်αါαα်။
The Graph of $\ y=a\sin\left(b\cdot\theta + c\right)+d$
Grade 11 αွα် αေ့αာαဲ့αြီးαြα ်αော Function Transformation ၏ rule αျားα‘αိုα်း $\ y=a\sin\left(b\cdot\theta + c\right)+d$ αွα် $c$ αα် $y=\sin\theta$ αို horizontal translation αြုαုα်αေးαြα်း αြα ်αြီး $d$ αα် $y=\sin\theta$ αို vertical translation αြုαုα်αေးαြα်း αြα ်αα်αု αိαှိααα် αြα ်αα်။
α₯ααာ α‘αေαြα့် $y=\sin\theta$ αှα့် $y=\displaystyle\frac{3}{2}\sin\left(2\theta-\frac{\pi}{3}\right)$ αို့αို
αှိုα်းαှα်αေ့αာαြα့်αါαα်။
For $y=\sin\theta$,
period = $2\pi$
amplitude = $1$
$\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & \displaystyle\frac{\pi}{2} & \pi & \displaystyle\frac{3\pi}{2} & 2\pi\\
\hline
y & 0 & 1 & 0 & -1 & 0\\
\hline
\end{array}$
For $y=\displaystyle\frac{3}{2}\sin\left(2\theta-\frac{\pi}{3}\right)$,
period = $\frac{1}{2}\times2\pi=\pi$
amplitude = $\displaystyle\frac{3}{2}$
Let $f(\theta)=\sin\theta$ then $y=\displaystyle\frac{3}{2}\sin\left(2\theta-\frac{\pi}{3}\right)
=\displaystyle\frac{3}{2}f\left(2\theta-\frac{\pi}{3}\right)$.
α‘αα်αါ ααားαွα် αော်αြαားαော $y=\sin\theta$ αေါ်αွα်αှိαော α‘αှα်αျား ၏
$y=\displaystyle\frac{3}{2}\sin\left(2\theta-\frac{\pi}{3}\right)$ αေါ်αွα်αှိαော mapped point αျားαို
transformation method αြα့်αှာαြα့်αါαα်။
$\begin{array}{l}
\displaystyle(0,0)\xrightarrow{f\left(\theta-\frac{\pi}{3}\right)}\left(\frac{\pi}{3},0\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{\pi}{6},0\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{\frac{3}{2}f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{\pi}{6},0\right)\\
\end{array}$
$\begin{array}{l}
\displaystyle(\frac{\pi}{2},1)\xrightarrow{f\left(\theta-\frac{\pi}{3}\right)}\left(\frac{5\pi}{6},1\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{5\pi}{12},1\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{\frac{3}{2}f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{5\pi}{12},\frac{3}{2}\right)\\
\end{array}$
$\begin{array}{l}
\displaystyle(\pi,0)\xrightarrow{f\left(\theta-\frac{\pi}{3}\right)}\left(\frac{4\pi}{3},0\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{2\pi}{3},0\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{\frac{3}{2}f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{2\pi}{3},0\right)\\
\end{array}$
$\begin{array}{l}
\displaystyle(\frac{3\pi}{2},-1)\xrightarrow{f\left(\theta-\frac{\pi}{3}\right)}\left(\frac{11\pi}{6},-1\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{11\pi}{12},-1\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{\frac{3}{2}f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{11\pi}{12},-\frac{3}{2}\right)\\
\end{array}$
$\begin{array}{l}
\displaystyle(2\pi,0)\xrightarrow{f\left(\theta-\frac{\pi}{3}\right)}\left(\frac{7\pi}{3},0\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{7\pi}{6},0\right)\\
\hspace{1.3cm}\displaystyle\xrightarrow{\frac{3}{2}f\left(2\theta-\frac{\pi}{3}\right)}\left(\frac{7\pi}{6},0\right)\\
\end{array}$
αို့αြောα့် $y=\sin\theta$ αှα့် $y=\displaystyle\frac{3}{2}\sin\left(2\theta-\frac{\pi}{3}\right)$ αို့၏
graph αှα
်αုαို sketch αုα်αိုα်αြီαြα
်αာ α‘ောα်αါα‘αိုα်း ααှိαα်αြα
်αα်။
The Graph of $\ y=\cos\theta $
$\ y=\cos\theta $ ၏ graph αα် αα်း sine wave αဲ့αို့αα် one complete cycle αြα ်αα် $\theta=360^{\circ} = 2\pi$ αြα ်αα်။ αို့αြောα့် $\ y=\cos\theta $ ၏ period αှာ $2\pi$ αြα ်αα်။ $\ y=\sin\theta $ αဲ့αို့αα် $\ y=\cos\theta $ ၏ range αα်αα်း $-1\le\cos\theta\le 1 $ αြα ်αာ amplitude αှာ 1 iunit αြα ်αα်။
$\displaystyle\sin \left(\frac{\pi}{2}-\theta=\cos\theta\right)$ αု αိαှိαဲ့αြီးαြα
်αα်။
$y=\cos\theta$ αα် $y=\sin\theta$ αို horizontal translation αြုαုα်αားαြα်းαα်αြα
်αα်။
αို့αြောα့် $y=\cos\theta$ αα် $y=\sin\theta$ ၏ αုα်αα္αိαျားαှα့် αူαီαα်။
sine function αှα့်
cosine function αျား၏ αုα်αα္αိαျားαို α‘ောα်αါ α‘αိုα်းα‘αျုα်αှα်αိုα်αα်။
| Functions | Amplitude | Period | Range |
| $y=\sin\theta$ | 1 | $2\pi$ | $-1\le\sin\theta\le 1$ |
| $y=a(b\cdot \sin\theta+c)+d$ | a | $\displaystyle\frac{2\pi}{b}$ | $-a+d\le a(b\cdot \sin\theta+c)+d\le a+d$ |
| $y=\cos\theta$ | 1 | $2\pi$ | $-1\le\sin\theta\le 1$ |
| $y=a(b\cdot \cos\theta+c)+d$ | a | $\displaystyle\frac{2\pi}{b}$ | $-a+d\le a(b\cdot \cos\theta+c)+d\le a+d$ |
Question 1
The diagram shows part of the curve with equation
$y=p \sin (q \theta)+r$, where $p, q$ and $r$ are constants.
(a) State the value of $p$.
(b) State the value of $q$.
(c) State the value of $r$.
Question 2
The diagram shows part of the graph of $y=a \cos (b x)+c$.
(a) Find the values of the positive integers $a, b$ and $c$.
(b) For these values of $a, b$ and $c$, use the given diagram to determine
the number of solutions in the interval $0 \leqslant x \leqslant 2 \pi$ for
each of the following equations.
(i) $\displaystyle a \cos (b x)+c=\frac{6}{\pi} x$
(ii) $\displaystyle a \cos (b x)+c=6-\frac{6}{\pi} x$.
For the graph $y=\displaystyle\frac{6}{\pi} x$Question 3
The diagram shows the graph of $y=f(x)$, where $f(x)=\displaystyle\frac{3}{2} \cos 2 x+\frac{1}{2}$
for $0 \leqslant x \leqslant \pi$.
(a) State the range of $f$.
A function $g$ is such that $g(x)=f(x)+k$, where $k$ is a positive constant.
The $x$-axis is a tangent to the curve $y=g(x)$.
(b) State the value of $k$ and hence describe fully the transformation that maps
the curve $y=f(x)$ on to $y=g(x)$.
$\therefore\quad k=1$
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