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Inverse Trigonometric
Functions
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Inverse Trigonometric
Functions
Definition of the Inverse Trig. Functions
1. arcsin x quadrants I and IV
22
π π ≤≤− y
2. arccos x quadrants I and II π ≤≤
y0
3. arctan x quadrants I and IV
22
π π
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Evaluate each of the following (remember
that the output of an inverse trigonometric
function is an angle)
What angle beteen and has a sine
value of !1"2# 2
π −
2
π
$ou should %no hat angle has a sine value of
!1"2.
−−
2
1sin
1
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62
1sin
1 π −=
−−
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Evaluate each of the following (remember
that the output of an inverse trigonometric
function is an angle)
)0(cos 1− What angle beteen & and ' has a cosinevalue
of
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2)0(cos 1 π =−
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What angle beteen and has a
tangent value of #
Evaluate each of the following (remember
that the output of an inverse trigonometric
function is an angle)
)3(tan 1− 2π
−2
π
3
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What angle beteen and has a
tangent value of #
Evaluate each of the following (remember
that the output of an inverse trigonometric
function is an angle)
)3(tan 1− 2π
−
3
1
2
3
2
1
6cos
6sin
6tan =
=
=
π
π
π
2
π
3
(hec% '")*
(hec% '"3*
3
2
1
2
3
3cos
3sin
3tan =
=
=
π
π
π
Thus 3)3(tan 1 π
=−
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What angle beteen and has a
sine value of #
Evaluate each of the following (remember
that the output of an inverse trigonometric
function is an angle)
−
2
2sin 1 2
π
− 2
π
2
2
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422sin 1 π =
−
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Evaluate each of the following (remember
that the output of an inverse trigonometric
function is an angle)
What angle beteen & and ' has a cosine
value of !1#)1(cos
1
−
−
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π =−− )1(cos 1
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Find the values of *
−
21arcsin I or IV
!12
3
630 π
−°− or
23sin
1− I or IV 32
13
60 π or °
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2
2arccos I or II
1
21
!
π or °
( )1arccos −
!1
1&
π or °1"0
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#omposition of Functions
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+o e ,lug this into the original
ex,ression re,lace the
To evaluate e as% the question
-What angle beteen '"2 and '"2 has a sine
value of 1#/ This is one that e should
%no.
Evaluate each of the following
( ))1(sinsin 1−
)1(sin 1−
2)1(sin 1
π
=−
The first ste, in evaluating this ex,ression is
to find the value of the inside ,arentheses.
)1(sin 1−
2
sin π
This e can evaluation to get 1. Thus ( ) 1)1(sinsin
1
=−
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To evaluate e as% the question -What
angle beteen & and ' has a cosine value of #/
This is one that e should %no.
Evaluate each of the following
2
3
−23cossin 1
−2
3cos 1
+o e ,lug this into the original ex,ression
re,lace the
The first ste, in evaluating this ex,ression is to find
the value of the inside ,arentheses.
This e can evaluation to get 1"2. Thus
62
3cos 1
π =
−
6
sin π
2
1
2
3cossin 1 =
−
−2
3cos 1
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Evaluate each of the following
( ))1(tancos 1−
4
)1(tan 1 π =−
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( )2
2)1(tancos 1 =−
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Evaluate each of the following
−45sinsin 1 π
2
2
4
5sin −=
π
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44
5sinsin 1
π π −=
−
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Evaluate each of the following
−
3
5coscos 1
π
2
1
3
5cos =
π
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33
5coscos 1
π π
=
−
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Evaluate each of the following
−4
3coscos
1 π
22
43cos −=
π
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4
3
2
2cos 1
π =
−−
4
3
4
3
coscos 1 π π
=
−
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Find the exact value.
32arccostan
2
3 !
2
!
== adjopp
−!
3arcsincos
!30
!
==
hyp
adj
