lesson 1- the trig functions

8/8/2019 Lesson 1- The Trig Functions

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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



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



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



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?



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



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



-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.



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



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



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



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

lesson 1- the trig functions

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