462
chapter 6 Applications of Trigonometry
This result implies
that a regular octagon
whose vertices are
equally spaced points
on the unit circle
has
√
perimeter 8 2 − 2.
each side of this regular octagon equals the distance between (1, 0) and
which equals
!
√ √ 2
2 2
+ 22 .
1− 2
√2
2
,
√ 2
,
2
Simplifying the expression above, we conclude that each side of this regular
octagon has length
√
2 − 2.
(b) The Area Stretch Theorem (see Section 4.2) implies that there is a constant c such
that a regular octagon with sides of length s has area cs 2 . From the previous
√
example
and from part (a) of this example, we know that the area equals 2 2 if
√
s = 2 − 2. Thus
√ 2
√
√
2 2 = c 2 − 2 = c(2 − 2).
Solving this equation for c, we have
√
√
√
√
2 2
2 2
2+ 2
√ =
√ ·
√ = 2 2 + 2.
c=
2− 2
2− 2 2+ 2
Thus a regular octagon with sides of length s has area
√
(2 2 + 2)s 2 .
Most coins are round, but a few countries have coins that are regular polygons. The picture in the margin shows the one-dollar Canadian coin, which
is an 11-sided regular polygon. The techniques used in the example above
will allow you to compute the area of a face of this coin, as you are asked to
do in Exercise 38.
exercises
1.
2.
3.
4.
Find the area of a triangle that has sides of
length 3 and 4, with an angle of 37◦ between
those sides.
a
Θ
Find the area of a triangle that has sides of
length 4 and 5, with an angle of 41◦ between
those sides.
Find the area of a triangle that has sides of
length 2 and 7, with an angle of 3 radians between those sides.
Find the area of a triangle that has sides of
length 5 and 6, with an angle of 2 radians between those sides.
For Exercises 5–12 use the following figure (which
is not drawn to scale):
b
5. Find the value of b if a = 3, θ = 30◦ , and the
area of the triangle equals 5.
6. Find the value of a if b = 5, θ = 45◦ , and the
area of the triangle equals 8.
7. Find the value of a if b = 7, θ =
of the triangle equals 10.
π
4
, and the area
8. Find the value of b if a = 9, θ =
of the triangle equals 4.
π
3
, and the area
9.
Find the value of θ (in radians) if a = 7,
b = 6, the area of the triangle equals 15, and
π
θ < 2.
section 6.1 Using Trigonometry to Compute Area
463
10.
Find the value of θ (in radians) if a = 5,
b = 4, the area of the triangle equals 3, and
π
θ < 2.
23.
Find the value of ν (in degrees) if a = 6,
b = 7, and the area of the parallelogram equals
31.
11.
Find the value of θ (in degrees) if a = 6,
b = 3, the area of the triangle equals 5, and
θ > 90◦ .
24.
Find the value of ν (in degrees) if a = 8,
b = 5, and the area of the parallelogram equals
12.
12.
Find the value of θ (in degrees) if a = 8,
b = 5, and the area of the triangle equals 12,
and θ > 90◦ .
25. What is the largest possible area for a triangle
that has one side of length 4 and one side of
length 7?
13.
Find the area of a parallelogram that has
pairs of sides of lengths 6 and 9, with an angle
of 81◦ between two of those sides.
26. What is the largest possible area for a parallelogram that has pairs of sides with lengths 5 and
9?
14.
Find the area of a parallelogram that has
pairs of sides of lengths 5 and 11, with an angle
of 28◦ between two of those sides.
27. Sketch the regular hexagon whose vertices are
six equally spaced points on the unit circle,
with one of the vertices at the point (1, 0).
15. Find the area of a parallelogram that has pairs
π
of sides of lengths 4 and 10, with an angle of 6
radians between two of those sides.
16. Find the area of a parallelogram that has pairs
π
of sides of lengths 3 and 12, with an angle of 3
radians between two of those sides.
For Exercises 17–24, use the following figure
(which is not drawn to scale except that u is indeed meant to be an acute angle and ν is indeed
meant to be an obtuse angle):
b
a
a
Ν
u
b
17. Find the value of b if a = 4, ν = 135◦ , and the
area of the parallelogram equals 7.
◦
18. Find the value of a if b = 6, ν = 120 , and the
area of the parallelogram equals 11.
π
19. Find the value of a if b = 10, u = 3 , and the
area of the parallelogram equals 7.
20. Find the value of b if a = 5, u =
of the parallelogram equals 9.
21.
22.
π
4
, and the area
Find the value of u (in radians) if a = 3,
b = 4, and the area of the parallelogram equals
10.
Find the value of u (in radians) if a = 4,
b = 6, and the area of the parallelogram equals
19.
28. Sketch the regular dodecagon whose vertices
are twelve equally spaced points on the unit circle, with one of the vertices at the point (1, 0).
[A dodecagon is a twelve-sided polygon.]
29. Find the coordinates of all six vertices of the
regular hexagon whose vertices are six equally
spaced points on the unit circle, with (1, 0) as
one of the vertices. List the vertices in counterclockwise order starting at (1, 0).
30. Find the coordinates of all twelve vertices of
the dodecagondodecagon whose vertices are
twelve equally spaced points on the unit circle, with (1, 0) as one of the vertices. List the
vertices in counterclockwise order starting at
(1, 0).
31. Find the area of a regular hexagon whose vertices are six equally spaced points on the unit
circle.
32. Find the area of a regular dodecagon whose
vertices are twelve equally spaced points on the
unit circle.
33. Find the perimeter of a regular hexagon whose
vertices are six equally spaced points on the
unit circle.
34. Find the perimeter of a regular dodecagondodecagon whose vertices are twelve equally
spaced points on the unit circle.
35. Find the area of a regular hexagon with sides of
length s.
36. Find the area of a regular dodecagon with sides
of length s.