# Double-Angle And Half-Angle Identities Worksheets With Answer Page 9

479
6-3 Double-Angle and Half-Angle Identities
In Problems 55–60, ﬁnd the exact value of each without using
(B) Using the resulting equation in part A, determine the
a calculator.
angle that will produce the maximum distance d for
a given initial speed v
. This result is an important
0
3
3
1
1
55.
cos [2 cos
(
)]
56.
sin [2 cos
(
)]
consideration for shot-putters, javelin throwers, and
5
5
4
3
discus throwers.
1
1
57.
tan [2 cos
(
)]
58.
tan [2 tan
(
)]
5
4
1
3
1
4
1
1
59.
cos [
cos
(
)]
60.
sin [
tan
(
)]
2
5
2
3
In Problems 61–66, graph f(x) in a graphing utility, ﬁnd a
simpler function g(x) that has the same graph as f(x), and
verify the identity f(x)
g(x). [Assume g(x)
k
A T(Bx)
where T(x) is one of the six trigonometric functions.]
61. f(x)
csc x
cot x
62. f(x)
csc x
cot x
70. Geometry. In part (a) of the ﬁgure, M and N are the mid-
1
2 cos 2x
1
2 cos 2x
points of the sides of a square. Find the exact value of
63.
f(x)
64.
f(x)
2 sin x
1
1
2 cos x
cos . [Hint: The solution uses the Pythagorean theorem,
the deﬁnition of sine and cosine, a half-angle identity, and
1
cot x
65.
f(x)
66.
f(x)
some auxiliary lines as drawn in part (b) of the ﬁgure.]
cot x sin 2x
1
1
cos 2x
M
M
APPLICATIONS
67. Indirect Measurement. Find the exact value of x in the
/2
s
s
N
N
ﬁgure; then ﬁnd x and to three decimal places. [Hint:
/2
2
Use cos 2
2 cos
1.]
s
s
(a)
(b)
x
7 m
8 m
71.
Area. An n-sided regular polygon is inscribed in a circle
(A) Show that the area of the n-sided polygon is given by
68. Indirect Measurement. Find the exact value of x in the
1
2
2
A
nR
sin
ﬁgure; then ﬁnd x and to three decimal places. [Hint:
n
2
n
2
Use tan 2
(2 tan )/(1
tan
).]
1
[Hint: (Area of a triangle)
( )(base)(altitude). Also,
2
a double-angle identity is useful.]
(B) For a circle of radius 1, complete Table 1, to ﬁve
decimal places, using the formula in part A:
x
T A B L E
1
n
10
100
1,000
10,000
4 feet
2 feet
A
n
— Physics. The theoretical distance d that a shot-
69. Sports—
(C) What number does A
seem to approach as n
putter, discus thrower, or javelin thrower can achieve on a
n
given throw is found in physics to be given approximately
increases without bound? (What is the area of a circle
by
2
2v
sin cos
(D) Will A
exactly equal the area of the circumscribed
0
n
d
circle for some sufﬁciently large n? How close can A
32 feet per second per second
n
be made to get to the area of the circumscribed circle?
where v
is the initial speed of the object thrown (in feet
0
[In calculus, the area of the circumscribed circle is
per second) and is the angle above the horizontal at
called the limit of A
as n increases without bound. In
which the object leaves the hand (see the ﬁgure).
n
symbols, for a circle of radius 1, we would write
(A) Write the formula in terms of sin 2 by using a
lim
A
. The limit concept is the cornerstone on
n
n→
suitable identity.
which calculus is constructed.]