Section 6.4 Graphs of the sine and cosine functions

Transcription

1 Section 6. Grphs of the sine nd cosine functions This is the grph of the sine function f() sin f() sin Domin All rel numbers (, ) Rnge [ 1,1] Amplitute 1 Period π This sine function hs Period of π mens tht the sine function repets its vlues ever π rdins or 6.The mplitude of 1 mens tht the grph of sine hs its -vlues rnge between 1 nd 1. This is the grph of the cosine function f() cos f() cos Domin All rel numbers (, ) Rnge [ 1,1] Amplitute 1 Period π

2 A. How to find the Amplitude nd Period using the Trig function For both the sine nd cosine functions there is simple formul for getting the Amplitude nd period. So if f() sin (ω) Then Amplitude Period T 6 ω if in degrees nd Period T π ω Emple 1 : Determine the Amplitude nd Period of the following Trig Functions. () f() 5sin() (b) 7cos ( π ) if the function is in rdins (c) f() cos(8) (d) 1sin ( π ) () f() 5sin() Amplitude 5 nd Period T 6 6 ω 18 (b) 7cos ( π ) Amplitude 7 nd Period T π ω π π π π rdins (c) f() cos(8) Amplitude nd Period T 6 6 ω 8 5 (d) 1sin ( π ) Amplitude 1 nd Period T π π π ω π π 8 rdins

3 B. How to find the formul of Trig Function from sketch of its grph. The grphs below re of trig functions with n Amplitude of nd period of p we cn find their formul f() sin(k) of f() cos(k) b simpl finding the vlues of nd k. Amplitude of the Trig Function nd k cn be found b using the formul 6 ω 6 or if we re using rdins or ω π π. period T period T T T This is the Grph of sine function f() sin(ω) This is the Grph of cosine function f() cos(ω) We cn now use these properties of the trig functions to find their formul from their grphs. Emple 1 : Find the for the Trig Functions Trig Function from the following grphs Grph 1 Grph This is sine function with n Amplitude of 5 nd Period of 18 So ω 6 18 So ω π 8 This is cosine function with n Amplitude of nd Period of 8 rdins π 8 π 8 π So f () 5 sin() So f() cos ( π )

4 Emple : Find the formul for the Trig Functions from the following grphs Grph 1 Grph This is sine function reflected This is cosine function reflected in the -is so we will hve in the -is so we will hve sin s our strting function cos s our strting function Amplitude of nd Amplitude of 6 nd Period of 1 Period of rdins So ω 6 1 So ω π π So f () sin() So f() 6 cos ( π ) C. How to mke simple sketch of Trig Function from its formul. The grphs below re of trig functions with n Amplitude of nd period of p we cn find their formul f() sin(k) of f() cos(k) b simpl finding the vlues of nd p. Amplitude of the Trig Function nd p cn be found b using the formul Period T 6 ω degrees or if we re using rdins or Period T π ω rdins. p p This is the Grph of sine function f() sin(k) This is the Grph of cosine function f() cos(k) We cn now use these properties of the trig functions to sketch their grphs.find their formul

5 Emple 1 : Sketch the following Trig Functions. So f () sin() So f() 5 cos ( π ) This is sine function with n This is cosine function with n Amplitude of nd Amplitude of 5 nd Period T 6 9 Period T π π π π π π These re the sketches of these functions Emple : Sketch the following Trig Functions. So f () 8 sin(1) So f() 7 cos ( π ) This is sine function reflected This is cosine function reflected in the -is in the -is Amplitude of 8 nd Amplitude of 7 nd Period T Period T π π π π These re the sketches of these functions π π Grph 1 Grph

6 D. How to find the Averge Rte of Chnge of Trig Function The Averge rte of Chn ge for n function f() from to b is given b the formul :- Averge f(b) f() b Emple 1 : Find the Averge Rte of Chnge of the function f() cos from to π Averge f(b) f() b f(π) f() π cos π cos π 1 1 π π Emple : Find the Averge Rte of Chnge of the function f() sin ( ) from to π Averge f(b) f() b f(π) f() π sin π sin π E. How to find fog() nd gof() of Trig Functions (#81 to 8) Emple 1 : Find fog() nd gof() when f() nd g() sin fog() f (g() ) f(sin ) sin E. Applictions. gof() g (f()) g() sin () π π Emple 1 : If the current I in mpers, flowing through n c (lternting current) t time t seconds is I(t) 1 sin(πt ) Wht is its Amplitude nd period nd drw the grph over two periods. f(t) 1 sin(πt ) Amplitude 1 π Period 1 sec π Grph of I(t) 1 sin(πt ) is given below Current I Time t First Ccle Second Ccle