θ
)
5. (5 pts.)
Establish the following identity.
Show all steps
very, very carefully.
1  sin(
α
)
cos(
α
)
=
cos(
α
)
1 + sin(
α
)
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6. (5 pts.) Obtain the exact value of cos(
π
/8).
Show clearly and
neatly all the uses of appropriate identities.
cos(
π
/8) =
7. (5 pts.) If csc(
θ
) = 4 and cos(
θ
) < 0, what is the exact value
of sin(2
θ
)?
?
Show clearly and neatly all your uses of
appropriate identities.
sin(2
θ
)=
8. (5 pts.) Express the following product as a sum containing
only sines or cosines.
sin(5
θ
)cos(3
θ
)=
9. (10 pts.)
Find the exact value of each of the following
expressions if tan(
α
) = 5/12 with
π
/2 <
α
<
π
and sin(
β
) = 1/2
with
π
<
β
<3
π
/2.
Show all your uses of appropriate identities.
sin(
α

β
)=
cos(
α
+
β
TEST02/MAC1114
Page 4 of 4
10. (20 pts.)
Time to pay the piper.
... Give the exact values
for the following:
(a)
sin(0)
=
(b)
sin(
π
/6)
=
(c)
sin(
π
/4)
=
(d)
sin(
π
/3)
=
(e)
sin(
π
/2)
=
(f)
cos(0)
=
(g)
cos(
π
/6)
=
(h)
cos(
π
/4)
=
(i)
cos(
π
/3)
=
(j)
cos(
π
/2)
=
11. (10 pts.)
In order to get a neat identity for
cos(
α
) + cos(
β
), one begins with the identity
(*)
cos(x + y) + cos(x  y) = 2 cos(x)cos(y)
a
n
ds
e
t
sx+y=
α
a
n
dxy=
β
in the left side of the
identity.
To make the substitution uniform, it is necessary to
replace the "x" and "y" on the right side of (*) with what they
are in terms of "
α
" and "
β
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 Spring '08
 Storfer
 Cos, Elementary algebra

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