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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

of radius R.

(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

of radius 1?)

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.]

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