Trigonometric Applications
1. An isosceles triangle has equal sides of 12.4 and an included angle of
93.4°. What is the length of the third side of the triangle to the nearest
tenth?
(1)
(2)
4.9
18.0
(3) 48.9
(4) 18.5
1
2. In triangle ABC, cos A= - , b = 6.5, and c= 7.2. The length of a to the
nearest hundredth is
(1)
(2)
11.87
8.87
2
(3) 7.87
(4) 6.78
3. In triangle MNP, m<MNP =120, NM= 5.8 centimeters, and NP=8.3
centimeters. The length of MP to the nearest tenth is which of the
following?
(1)
(2)
7.4 cm
11.3 cm
(3) 12.3 cm
(4) 13.3 cm
4. Three sides of a triangle measure 5,8, and 12. The triangle is
(1)
(2)
Isosceles
Right
(3) acute
(4) obtuse
5. A triangle has sides of lengths 7.1, 9.4, and 15.3. Which of the following
is the measure of the largest angle of the triangle?
(1)
(2)
135.6°
115.8°
(3) 103.6°
(4) 25.5°
6. In
KLM, k=9 centimeters, l =40 centimeters, and m= 41
centimeters. What is the measure of the largest angle of the triangle?
(1)
(2)
107°
90°
(3) 84°
(4) 79°
7. In
PQR, m<RPQ =131, q= 10.8 inches, and r=8.1 inches. What is the
length of side p to the nearest tenth?
(1)
(2)
17.2
14.7
(3) 12.9
(4) 11.5
8. Each base angle of isosceles triangle GHI measures 57.4°, while equal
̅̅̅̅ and 𝐻𝐼
̅̅̅ and 𝐻𝐼
̅̅̅̅ each measure 8.94 inches. The length of 𝐺𝐼
̅̅̅̅
sides 𝐺𝐻
each measure 8.94 inches. The length of ̅̅̅
𝐺𝐼 to the nearest hundredth of
an inch is
(1)
(2)
7.12
8.26
(3) 8.59
(4) 9.63
*9. In parallelogram ABCD, AB =11 inches and BC= 17 inches. If m<ABC =
102°36’, the length of diagonal ̅̅̅̅
𝐴𝐶 to the nearest tenth of an inch is
(1)
(2)
18.1
21.2
(3) 22.2
(4) 28.0
*10. Dimitri and Anna are in charge of setting the route for the Daffy Drivers’
Bike Race at the Country Fair. This is an event for children ages 8-14 in which
each biker must complete the triangular course and collect souvenirs along
the way. The distance from the start to the merry- go –round to the middle
school field is 2.9 miles, and the angle included between them is 51°24’. Find
the total distance, to the nearest hundredth of a mile, covered by the bikers in
this event.
START
2.9 miles
Middle School Field
51°24
’
Merry – Go- Round
11. Two cabins are situated on Lake Happy Trails, at a distance of 800 feet
apart. The owners of the cabins, the Browns and the Adlers, would like to put
a wooden raft out on the lake, so that it is 1,000 feet from each of the two
cabins, labeled A and B.
R
800
a)What is the measure of < RAB, to the nearest tenth of a degree?
b) What is the measure of < ARB, to the nearest tenth of a degree?
*12. The lighthouse of Pine Island is visible from two boats off shore. Doug’s
sailboat is 4.2 miles from the lighthouse while Ralph’s fishing trawler is 6.7
miles from the lighthouse. If the light from the lighthouse sweeps an angle of
66.5° between the two boats, how far apart are they, to the nearest tenth of a
mile?
13. Find the measure of angle BAC.
8.9
Find the indicated side to a to the nearest tenth or angle A to the nearest
tenth of a degree for each triangle.
14.
A
22.4°
22.4°
B
a
C
15.
A
33.4
B
101.2
a
C
16. In
DAY, sin D= 0.6437, sin A =0.8134, and a =13.2. The length of
d to the nearest tenth is
(1) 10.4
(3) 43.8
(2) 18.6
(4) 67.7
17. In an isosceles triangle, the base angles each measure 61° and the length of
each congruent leg is 12.5. Which of the following equations can be used to
find the length of the base?
(1)
𝑠𝑖𝑛52°
(2)
𝑠𝑖𝑛61°
(3)
𝑠𝑖𝑛122°
12.5
12.5
12.5
=
𝑠𝑖𝑛61°
=
𝑠𝑖𝑛58°
𝑥
=
𝑥
𝑠𝑖𝑛61°
𝑥
(4) x2 = (12.5)2 + (12.5)2-2(12.5)(12.5)sin61°
18. In
VAL, v =13.12, a =11.3, and m< A= 44.5. The triangle must be
which of the following?
(1) cannot be determined
(2) obtuse
(3) isosceles
(4) right
For 19-20 find the area of each triangle to the nearest hundredth of a unit.
19. In
ABC, m< A= 58.2, b =8.6 centimeters, and c=7.1 centimeters.
20. In isosceles
centimeters.
LUV, the measure of vertex angle U= 54° and v = 18
In 21-24, select the numeral preceding the expression that best completes the
sentence or answers the question.
21. If the area of
END is 24 square inches, m< E =150, and d
measures 12 inches, the length of side n is how many inches?
(1) 8
(3) 16
(2) 12
(4) 24 √3
22. If one side of an equilateral triangle measures 6, what is the exact area of
the triangle?
(1) 9
(3) 9 √3
(2) 6 √3
(4) 18
23. In
JOG, m < JOG = 82, j=8.4, and g= 7.1. The area of
JOG is approximately:
(1) 14.76
(3) 59.06
(2) 29.53
(4) 59.64
*24. In parallelogram JACK, <JCK measures 61°15’, JC=23.5, and KC=18.7.
The area of the parallelogram is approximately which of the following?
(1) 192.64
(3) 423.18
(2) 385.28
(4) 462.11
25. Given parallelogram TIME as shown in the diagram. The lengths, in
centimeters, of the sides of triangle TIE are TI= 16.4, TE= 15.8, and EI= 21.3.
16.4 cm
I
T
15.8 cm
M
E
a) Find the measure of < TIE to the nearest tenth of a degree.
b) Find the area of parallelogram TIME to the nearest tenth of a square
centimeter.
26. A local airline does not offer direct connection from city A to city B.
Rather, the flight travels 40 miles from city A to city C, then 70 miles from C to
B. If m <ACB =110°, find the distance between city A and city B to the nearest
mile.
1
27. In
ABC, a=6, b=8, and sin C= . Find the area of
4
ABC.
28. Two sides of a parallelogram are 5 and 7 and the included angle is 60°.
Find the length of the shorter diagonal to the nearest tenth.
29. Find the area of triangle ABC, to the nearest integer, if a=6, b=8, and m
<C=133°
Two consecutive sides of a parallelogram are 6 cm and 4 cm.
30. If the length of the longer diagonal of the parallelogram is 9 centimeters,
find the measure of the largest angle of the parallelogram to the nearest
degree.
31. In triangle ABC, m<A=47°, a=50, b=63, and angle B is obtuse. What is
m<C to the nearest degree?
32. In triangle ABC, m<A=47°, a=50, b=63, and angle B is obtuse. What is
m<C to the nearest degree?
33. In triangle ABC, m<A=35°, m<B=48°, and a=16. What is the measure of c
to the nearest tenth?
*34. A triangular plot of land has sides that measure 5 meters, 7 meters, and
10 meters. What is the area of this plot of land? (HINT: Find an angle first!)