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Trig Functions
Learning Objectives:Learning Objectives:
Recognising sin, cos and tanRecognising sin, cos and tan
Understand how graphs of trig functions are transformed.Understand how graphs of trig functions are transformed.
Range and domainRange and domain
InversesInverses
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y = f(xy = f(x -- a)a)
yy -- b = f(x) b = f(x)
yy -- b = f(x b = f(x -- a)a)
y = k f(x)y = k f(x)
y =y = -- f(x)f(x)
stretches by a factor µk¶ along the ystretches by a factor µk¶ along the y--axisaxis
y = f(x/k)y = f(x/k) stretches by a factor µk¶ along the xstretches by a factor µk¶ along the x--axisaxis
y = f(y = f(--x)x)
reflects in the xreflects in the x--axisaxis
reflects in the yreflects in the y--axisaxis
[ ][ ]aa
b b
Translation by
[ ][ ]00
b bTranslation by
[ ][ ]aa
00Translation by
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Trig Graphs: translation
How will y=sin x, y=sin x +1 y=sin xHow will y=sin x, y=sin x +1 y=sin x ± ± 33
work in degrees.work in degrees.
How is the y=sin x transformed to makeHow is the y=sin x transformed to make
the other two graphs?the other two graphs?
y = sin x
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
x
y
y = sin x
y = sin x -
y = x - 0
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y = s x
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0 45 90 135 180 225 270 315 360
x
y
y s n
y s n 1
y - 3
y = s n x - 3
--For y=s n x + 1 there is a translation of 1 unit up. (For y=sin x + 1 there is a translation of 1 unit up. ( y y -- 1 = sin x1 = sin x ))
--For y=sin xFor y=sin x -- 3 there is a translation of 3 there is a translation of --3 unit up. (3 unit up. ( y + 3 = sin xy + 3 = sin x ))
--What about y=cos x + 2 or y=tanxWhat about y=cos x + 2 or y=tanx ± ± 4?4?
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Graphs: more translation
Plot using a graphic calculator and then sketchPlot using a graphic calculator and then sketchy=sin x,y=sin x, y=sin( x + 90) y=sin( x + 90) and y=sin(xand y=sin(x--45).45).
How is the y=sin x transformed to make theseHow is the y=sin x transformed to make these
two graphs?two graphs?
y = sin x
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
x
y
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Graphs: Cosine
Plot using a graphic calculator and thenPlot using a graphic calculator and thensketch y=cos x, y=2 cos x and y=cos 1/4x.sketch y=cos x, y=2 cos x and y=cos 1/4x.
How is theHow is the y=cos xy=cos x transformed to maketransformed to make
these two graphs?these two graphs?y = sin x
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0 90 180 270 360
x
y
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-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.01.5
2.0
2.5
0 90 180 270 360
x
y
=
= 2 ( )
= ( /4)
For =2 os there is a stret h of 2 in the dire tion.For =2 os there is a stret h of 2 in the dire tion.
For = os 1/4 there is a stret h of 4 in the dire tion.For = os 1/4 there is a stret h of 4 in the dire tion.
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Range and Domain (1)The 3 T g Function
-
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0 45 90 135 180 225 270 315 360
x
y = sin x
y = cos x
y= tan x
Tr ig functions go on f orever, so the domain has to be specif iedTr ig functions go on f orever, so the domain has to be specif ied
Here the domain is 0Here the domain is 0 x 360 x 360
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Range and Domain (2)The 3 T ig Function
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
0 45 90 135 180 225 270 315 360
x
y = sin x
y = cos x
y= tan x
00 x 360 x 360
The r ange?The r ange?
sin x and cos x have a maximum of 1sin x and cos x have a maximum of 1
sin x and cos x have a minimum of sin x and cos x have a minimum of --11
--11 sin x 1 sin x 1
--11 cos x 1 cos x 1
tantan x is unbound x is unbound
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Inverse Functions (2)
As always in maths, there is a trick to this« As always in maths, there is a trick to this«
1.1. Write function as a rule in terms of y and x.Write function as a rule in terms of y and x.
2.2. Swap µx¶ and µy¶Swap µx¶ and µy¶
3.3. Rearrange to get in terms of y.Rearrange to get in terms of y.
f(x) = sin x
y = sin x
x = sin y
y = sin-1 x
f -1
(x) = sin-1
x
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Inverse Functions (1)
f(x) = sin x f -1(x) = sin-1 x
FunctionFunction InverseInverse
f(x) = cos x f -1(x) = cos-1 x
f(x) = tan x f -1(x) = tan-1 x
These have always been calledThese have always been called inverse functionsinverse functions
and the symbol (and the symbol (--11) is the same) is the same
± ± so it is kind of obviousso it is kind of obvious