Menu Close

State-the-phase-shift-the-amplitude-and-draw-the-graph-a-g-3-4-sin-2-pi-b-f-1-3-4-sin-2-pi-c-f-4-




Question Number 42036 by Tawa1 last updated on 17/Aug/18
State the phase shift,  the amplitude and draw the graph.  (a)  g(θ) = (3/4) sin(2θ + π)  (b)  f(θ) = 1 + (3/4) sin(2θ + π)  (c)  f(θ) = 4θ
$$\mathrm{State}\:\mathrm{the}\:\mathrm{phase}\:\mathrm{shift},\:\:\mathrm{the}\:\mathrm{amplitude}\:\mathrm{and}\:\mathrm{draw}\:\mathrm{the}\:\mathrm{graph}. \\ $$$$\left(\mathrm{a}\right)\:\:\mathrm{g}\left(\theta\right)\:=\:\frac{\mathrm{3}}{\mathrm{4}}\:\mathrm{sin}\left(\mathrm{2}\theta\:+\:\pi\right) \\ $$$$\left(\mathrm{b}\right)\:\:\mathrm{f}\left(\theta\right)\:=\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:\mathrm{sin}\left(\mathrm{2}\theta\:+\:\pi\right) \\ $$$$\left(\mathrm{c}\right)\:\:\mathrm{f}\left(\theta\right)\:=\:\mathrm{4}\theta \\ $$$$ \\ $$
Commented by maxmathsup by imad last updated on 17/Aug/18
a) g(θ) =−(3/4)sin(2θ)    we have g(θ+π) =−(3/4)sin(2(θ+π))  =−(3/4)sin(2θ +2π) =−(3/4)sin(2θ) =g(θ) ⇒ g periodic with T=π   we can study the variation on [−(π/2),(π/2)]   but  g(−θ) =(3/4)sin(2θ)=−g(θ) ⇒  g is odd  so we can study the variation on [0,(π/2)] =[0,(π/4)]∪[(π/4),(π/2)]  g^′ (θ) =−(3/2)cos(2θ)    we have  2θ ∈[0,π]   x          0              (π/4)              (π/2)                        we have g(o)=0     , g((π/4))=−(3/4)  g^′ (x)        −                 +                                       g((π/2)) =0   ....  g(x)        decr          inc
$$\left.{a}\right)\:{g}\left(\theta\right)\:=−\frac{\mathrm{3}}{\mathrm{4}}{sin}\left(\mathrm{2}\theta\right)\:\:\:\:{we}\:{have}\:{g}\left(\theta+\pi\right)\:=−\frac{\mathrm{3}}{\mathrm{4}}{sin}\left(\mathrm{2}\left(\theta+\pi\right)\right) \\ $$$$=−\frac{\mathrm{3}}{\mathrm{4}}{sin}\left(\mathrm{2}\theta\:+\mathrm{2}\pi\right)\:=−\frac{\mathrm{3}}{\mathrm{4}}{sin}\left(\mathrm{2}\theta\right)\:={g}\left(\theta\right)\:\Rightarrow\:{g}\:{periodic}\:{with}\:{T}=\pi\: \\ $$$${we}\:{can}\:{study}\:{the}\:{variation}\:{on}\:\left[−\frac{\pi}{\mathrm{2}},\frac{\pi}{\mathrm{2}}\right]\:\:\:{but}\:\:{g}\left(−\theta\right)\:=\frac{\mathrm{3}}{\mathrm{4}}{sin}\left(\mathrm{2}\theta\right)=−{g}\left(\theta\right)\:\Rightarrow \\ $$$${g}\:{is}\:{odd}\:\:{so}\:{we}\:{can}\:{study}\:{the}\:{variation}\:{on}\:\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right]\:=\left[\mathrm{0},\frac{\pi}{\mathrm{4}}\right]\cup\left[\frac{\pi}{\mathrm{4}},\frac{\pi}{\mathrm{2}}\right] \\ $$$${g}^{'} \left(\theta\right)\:=−\frac{\mathrm{3}}{\mathrm{2}}{cos}\left(\mathrm{2}\theta\right)\:\:\:\:{we}\:{have}\:\:\mathrm{2}\theta\:\in\left[\mathrm{0},\pi\right]\: \\ $$$${x}\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\pi}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\pi}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{we}\:{have}\:{g}\left({o}\right)=\mathrm{0}\:\:\:\:\:,\:{g}\left(\frac{\pi}{\mathrm{4}}\right)=−\frac{\mathrm{3}}{\mathrm{4}} \\ $$$${g}^{'} \left({x}\right)\:\:\:\:\:\:\:\:−\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{g}\left(\frac{\pi}{\mathrm{2}}\right)\:=\mathrm{0}\:\:\:…. \\ $$$${g}\left({x}\right)\:\:\:\:\:\:\:\:{decr}\:\:\:\:\:\:\:\:\:\:{inc} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
steps to draw...  1)draw sinθ  in between  θ= 0 and 2Π...we know  the shape of sinθ then repeate the shape but  −ve side..  2)f(θ) 0   30^o   45^o   60^o   90^o   120^o   135^o   150^o  180^o        sinθ 0   (1/2)   (1/( (√2) ))  (((√3) )/2)   1     (((√3) )/2)    (1/( (√2) ))      (1/2)     0  now i attaching  to clarify...
$${steps}\:{to}\:{draw}… \\ $$$$\left.\mathrm{1}\right){draw}\:{sin}\theta\:\:{in}\:{between}\:\:\theta=\:\mathrm{0}\:{and}\:\mathrm{2}\Pi…{we}\:{know} \\ $$$${the}\:{shape}\:{of}\:{sin}\theta\:{then}\:{repeate}\:{the}\:{shape}\:{but} \\ $$$$−{ve}\:{side}.. \\ $$$$\left.\mathrm{2}\right){f}\left(\theta\right)\:\mathrm{0}\:\:\:\mathrm{30}^{{o}} \:\:\mathrm{45}^{{o}} \:\:\mathrm{60}^{{o}} \:\:\mathrm{90}^{{o}} \:\:\mathrm{120}^{{o}} \:\:\mathrm{135}^{{o}} \:\:\mathrm{150}^{{o}} \:\mathrm{180}^{{o}} \\ $$$$\:\:\:\:\:{sin}\theta\:\mathrm{0}\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:}\:\:\frac{\sqrt{\mathrm{3}}\:}{\mathrm{2}}\:\:\:\mathrm{1}\:\:\:\:\:\frac{\sqrt{\mathrm{3}}\:}{\mathrm{2}}\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:}\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\mathrm{0} \\ $$$${now}\:{i}\:{attaching}\:\:{to}\:{clarify}… \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
Commented by Tawa1 last updated on 17/Aug/18
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by Tawa1 last updated on 17/Aug/18
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by Tawa1 last updated on 17/Aug/18
I appreciate your effort
$$\mathrm{I}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{effort} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 17/Aug/18
if you wantfree books pdf form download able  seach archives.org then in search option write  physics or mathematics or graph or algebra  here you get irodov krotov sl loneh   algebra hall and knight etc...
$${if}\:{you}\:{wantfree}\:{books}\:{pdf}\:{form}\:{download}\:{able} \\ $$$${seach}\:{archives}.{org}\:{then}\:{in}\:{search}\:{option}\:{write} \\ $$$${physics}\:{or}\:{mathematics}\:{or}\:{graph}\:{or}\:{algebra} \\ $$$${here}\:{you}\:{get}\:{irodov}\:{krotov}\:{sl}\:{loneh}\: \\ $$$${algebra}\:{hall}\:{and}\:{knight}\:{etc}… \\ $$
Commented by Tawa1 last updated on 17/Aug/18
Wow, i will try it sir. God bless you sir
$$\mathrm{Wow},\:\mathrm{i}\:\mathrm{will}\:\mathrm{try}\:\mathrm{it}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}\: \\ $$
Commented by Tawa1 last updated on 17/Aug/18
It is good sir. thanks. God bless you
$$\mathrm{It}\:\mathrm{is}\:\mathrm{good}\:\mathrm{sir}.\:\mathrm{thanks}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *