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INSTRUCTOR: Dr. Bathi Kasturiarachi
NAME:
Math 11022
Spring 2008
Trigonometry
REVIEW for Exam 2 { Sections (6.5, 7.1, 7.2, 7.3)
1.
Find the exact value of
arcsin(1)
=
2.
2
Find the exact value of
tan
=
3.
1
tan
5
4
4
Find the exact value of
tan sin
1
1
2
1
1
, sin y = =) y =
2
2
p 6
1
3
) tan sin 1
= tan =
2
6
3
Let y = sin
4.
1
Find the exact value of
tan cos
1
2
5
2
2
, cos y = =) draw a triangle
5
5
p
21
1 2
= tan y =
) tan cos
5
2
Let y = cos
5.
1
Find an algebraic expression in terms of x for the following.
sin (arccos x)
x
=) draw a triangle
1 p
) sin (arccos x) = sin y = 1 x2
Let y = cos
6.
1
x , cos y =
Find an algebraic expression in terms of x for the following.
!
p
2
x
5
sec cos 1
2x
p
p
x2 5
x2 5
1
Let y = cos
, cos y =
=) draw a triangle
2x
2x
!
p
x2 5
2x
sec cos 1
= sec y = p
2x
x2 5
1
7.
Prove the following trigonometric identity.
1
1
1
+
cos x 1 + cos x
2 csc2 x
LHS =
=
8.
1 + cos x + 1 cos x
(1 cos x)(1 + cos x)
2
2
= 2 csc2 x = RHS
=
1 cos2 x
sin2 x
Prove the following trigonometric identity.
sin3 + cos3
(sin + cos )(1
sin cos )
2
LHS = (sin + cos )[sin
= (sin + cos )(1
9.
sin cos + cos2 ]
sin cos ) = RHS
Prove the following trigonometric identity.
tan2 x
sec x + 1
cos x
cos x
sec2 x 1
(sec x 1)(sec x + 1)
LHS =
=
sec x + 1
(sec x + 1)
1 cos x
1
1=
= RHS
= sec x 1 =
cos x
cos x
10.
Prove the following trigonometric identity.
(tan
sec )2
LHS =
=
=
11.
1
1 sin
1 + sin
2
1
sin
cos
cos
(1 sin )2
cos2
(1 sin )2
(1 sin )2
1 sin
=
=
2
(1
sin
)(1
+
sin
)
1 + sin
1 sin
Find exact value of sin(195o ).
sin(195o ) = sin(150 + 45 )
= sin 150 cos 45 + cos 150 sin 45
p
p p
p
p
1
2
3
2
2
6
=
=
2 2
2
2
4
12.
Find exact value of cos(17 =12).
cos(17 =12) = sin(255 ) = sin(210 + 45 )
= sin 210 cos 45 + cos 210 sin 45
p
p p
p
p
1
2
3
2
2
6
=
=
2
2
2
2
4
2
= RHS
13.
If sin =
1
5
nd the exact values of the following.
sin(
=4) = sin cos =4 cos sin =4
p "
p #
p
2
2 1
24
[sin
cos ] =
=
2
2 5
5
p
p
2(1
24)
=
10
cos(
=3) = cos cos =3 + sin sin =3
p
p
p
3
24 1 1
3
1
+ sin
=
+
= cos
2
2
5
2
5
2
p
p
( 24 + 3)
=
10
tan + tan =6
tan( + =6) =
1 tan tan =6
=
1
14.
p
p1 + 3
3
24
p
3 p1
3
24
Find the exact value of
sin sin
1
4
5
tan
1
Let y = sin
1
Let z = tan
1
3
4
= sin(y
4
5
1
sin sin
4
5
3
4
tan
, sin y =
, tan z =
1
3
4
4
=) draw a triangle in quad IV
5
3
=) draw a triangle in quad I
4
z)
= sin y cos z cos y sin z
4 4 3 3
= 1
=
5 5 5 5
15. Find the exact value of the expression:
sin 105o
= sin(60 + 45 )
= sin 60 cos 45 + cos 60 sin 45
p p
p
p
p
3
2 1
2
6+ 2
=
+
=
2
2
2 2
4
16.
Find the exact value of the expression:
5
12
= sin 75
sin
= sin(45 + 30 )
= sin 45 cos 30 + cos 45 sin 30
p
p
6+ 2
=
4
3
17.
For the angles
18.
If sin =
4
5
and
and
shown in the gures, nd cos( + ). Refer to picture given in class.
lies in quadrant III, nd sin( =2).
sin( =2)
r
1 cos
= +
2
s
r
r
3
1
8
4
2
5
=
=
=
=p
2
10
5
5
19. If sin =
3
7
and
lies in quadrant II, nd cos( =2).
cos( =2)
r
1 + cos
= +
2
s
s
s
p
p
40
1+ 7
7
40
7
=
=
=
2
14
2
3
20. Find the exact value of sin(2x), if cos x =
p
2 10
14
and 0o < x < 90o .
sin(2x)
= 2 sin
px cos x p
5 2
4 5
= 2
=
3 3
9
21.
Find the exact value of cos(2x), if tan x =
5
4
and x lies in quadrant III.
cos(2x)
= cos2 x
4
p
=
41
22.
sin2 x
2
p
5
41
2
=
16
25
41
=
9
41
Verify the identity:
sin 4
4 sin( ) cos( ) cos(2 )
LHS = 2 sin(2 ) cos(2 )
= 2 2 sin cos cos(2 )
= 4 sin( ) cos( ) cos(2 ) = RHS
23.
Verify the identity:
cos 4x
8 cos4 x
LHS = 2 cos2 (2x)
2
= 2[2 cos x
4
= 2[4 cos x
4
= 8 cos x
4
8 cos2 x + 1
1
1]2
1
2
4 cos x + 1]
2
1
8 cos x + 1 = RHS
24.
Express the following product as a sum.
sin 6 sin 4
OMIT for this test.
25.
Express the following product as a sum.
cos 3 cos
OMIT for this test.
26.
Express the following sum as a product.
sin 5
sin 3
OMIT for this test.
27.
Express the following sum as a product.
cos 3 + cos 2
OMIT for this test.
28.
Prove the identity:
sin 4x + sin 8x
sin 4x sin 8x
OMIT for this test.
5
tan 6x