13 inverse trig functions.ppt
TRANSCRIPT
7/27/2019 13 Inverse Trig Functions.ppt
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13: Inverse Trig Functions
© Christine Crisp
“Teach A Level Maths”
Vol. 2: A2 Core Modules
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Inverse Trig Functions
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Module C3
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Inverse Trig Functions
,
50sin x
If we use a calculator to find x for
we use the inverse sin function, .)50(sin1
Our calculators give us one solution, the principal
value, although we know there are an infinite numberof solutions.
We will now look in detail at the inverse functions:
x x x 111
tancossin
, ,
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Inverse Trig Functions
The notation for the inverse of is)(x f )(1
x f
A Bit More on Notation
WARNING The notation for a reciprocal and aninverse are the same!
x 11x e.g. means the reciprocal of x ,
50sin1
(it means undo the sine function )
but means the inverse of x sin
For the reciprocal of we must write .x sin 1sin
x
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Inverse Trig Functions
x y sin
What domain would you use?
We need to make sure that we have all the y -valueswithout any being repeated.
9090 x ( or, in degrees, as ).
The domain is defined as22
x
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Inverse Trig Functions
x y sin
To make it easier to sketch the inverse by reflectingin y = x , I’ll use the same scales on both axes.
5712
571
2
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Inverse Trig Functions
To make it easier to sketch the inverse by reflectingin y = x , I’ll use the same scales on the axes.
5712
x y sin
x y
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Inverse Trig Functions
To make it easier to sketch the inverse by reflectingin y = x , I’ll use the same scales on the axes.
5712
x y sin
x y x y 1
sin
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Inverse Trig Functions
x y 1
sin
The domain of an inverse function, , has the
same values as the range of and vice versa.
)(1
x f
)(x f 11 x So, the domain of isx y
1sin
and the range is
22
y
Domain and Range
1 1
2
2
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Inverse Trig Functions
x y 1sin
Finally, we can redraw using labels onthe axes that clearly show the domain and range.
x y 1
sin
11 x
22
y
It’s important to notice ( for later ) that thegradient is always positive except at whereit is undefined.
1x
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Inverse Trig Functions
Exercise
1. (a) Sketch for a suitable domain inorder to define its inverse.
(b) Sketch the inverse.
(c) Write down the domain and range.
(d) Write down what you notice about the gradient.
x y 1cos
x y 1
tan2. Repeat question 1. for .
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Inverse Trig Functions
x y 1
tan2
y
2
y
x y tan
x y
x y tan
x y
x y 1
tan
Solution: 2 or, using the usual scales,
(c) The domain of isx y 1
tan
and the range is
22
y
This is a strict inequality because of the asymptotes
x
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Inverse Trig Functions
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Inverse Trig Functions
The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and
photocopied. For most purposes the slides can be printedas “Handouts” with up to 6 slides per sheet.
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Inverse Trig Functions
11 x 22
y
It’s important to notice ( for later ) that thegradient is always positive except at whereit is undefined.
1x
Domain:
x y 1
sin
Range:
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Inverse Trig Functions
11 x y 0
The gradient is always negative except atwhere it is undefined.
1x
Domain: Range:
x y 1
cos
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Inverse Trig Functions
22
y x
The gradient is always 0
Domain: Range:
x y 1tan