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Regents Review: Trigonometric Applications
Area of a Triangle: A 
1
ab sin C
2
**a and b are sides and C is the included angle**
If two sides of a triangle are 16 cm and 12 cm and the included angle is 30 .
In the parallelogram HOPE, HO  16.7cm , HE  20.1cm , and mOHE  108.3 .
a. Find the length of OE to the nearest tenth of a centimeter
b. Find the area of parallelogram HOPE to the nearest hundredth of a square centimeter
Example Regents Questions
1.
In ABC , mA  120, b  10, c  18 . What is the area of ABC to the nearest square inch?
2.
(1) 52
(2) 78
(3) 90
(4) 156
The sides of a parallelogram measure 10 cm and 18 cm. One angle of the parallelogram measures 46 degrees.
What is the area of the parallelogram, to the nearest square centimeter?
(1) 65
3.
(2) 125
(3) 129
Determine the area, to the nearest integer, of SRO shown below.
(4) 162
Law of Cosines: when given two sides and the included angle, and are looking for the third side.
c 2  a 2  b 2  2ab cos C
In HAT , a  6.4 , t  10.2 , and mH  87 . Find the length of side h to the nearest tenth.
Jed is working on a stained glass project and needs to form a triangle with sides of 8, 12, and 15 inches out of lead cane
to enclose the glass. To the nearest tenth, what is the largest angle he needs to create using lead cane?
Example Regents Questions
1.
In ABC , a  3, b  5, and c  7 . What is mC ?
(1) 22
2.
(2) 38
(3) 60
(4) 120
In ABC , a  15, b  14, and c=13 , as shown in the accompanying diagram. What is the mC , to the
nearest degree?
(1) 53
(2) 59
(3) 67
(4) 127
3.
In isosceles triangle QRS, q  s  4.7cm . If cos R  0.1908 , find the length of side r, to the nearest hundredth
of a centimeter.
Law of Sines: use law of sines when you can setup a proportion.
a
b
c


sin A sin B sin C
In triangle CTH, mT  107.3 , mH  34.5 , and CH  17.2cm . Find the length of CT to the nearest tenth of a
centimeter.
In triangle PQR, mPQR  66.5 , mQRP  47 , and QR  12.4in . Find the length of QP to the nearest tenth of an
inch.
The ambiguous case: The situation described above is known as the ambiguous case. When the given information is
given as Side-Side-Angle (SSA) or (ASS) there is not sufficient information to determine the side and shape of the triangle
based on solving for an angle.
# of possible triangle: 0, 1, or 2.
Given triangle JKL in which JK  14.3 , KL  5.8 , and mJ  17 , find the measure of  L to the nearest tenth of a
degree.
Determine the number of triangle given a  8 , b  10 , and mA  20 .
Example Regents Questions
1.
In ABC , mA  74, a  59.2, and c  60.3 . What are the two possible values for mC , to the nearest
tenth?
(1) 73.7 and 106.3
(2) 73.7 and 163.7
2.
In ABC , a  3, b  5, and c  7 . What is mC ?
(2) 22
3.
(2) 38
(3) 60
(4) 120
How many distinct triangles can be formed if mA  35, a  10, and b=13 ?
(1) 1
4.
(3) 78.3 and 101.7
(4) 78.3 and 168.3
(2) 2
(3) 3
(4) 0
In ABC , mA  32, a  12, and b  10 . Find the measure of the missing angles and side of ABC .
Round each measure to the nearest tenth.
Mixed Applications—Example Regents Questions
1.
In a triangle, two sides that measure 6 cm and 10 cm form an angle that measures 80 . Find, to the nearest
degree, the measure of the smallest angle in the triangle.
2. The diagram below shows the plans for a cell phone tower. A guy wire attached to the top of the tower makes
an angle of 65 degrees with the ground. From a point on the ground 100 feet from the end of the guy wire, the
angle of elevation to the top of the tower is 32 degrees. Find the height of the tower, to the nearest foot.
Forces and Vectors: two forces acting on an object can be represented by vectors. Vectors are very useful in
representing the properties of applied forces pushing or pulling an object in a given direction. The new force, created by
the combination of the initial forces, is called the resultant.
In force problems, each vector represents a force applied to an object at a specific point. These ectors are used to form
two consecutive sides of a parallelogram with the resultant force of the vectors forming the diagonal of the
parallelogram.
Example:
Two forces of 28 pounds and 41 pounds act on a body so that the angle between the two forces measures 72 degrees.
Find, to the nearest tenth of a pound, the magnitude of the resultant the forces produce.
A resultant force of 162 pounds must be exerted to move a refrigerator. If the two applied forces act on the refrigerator
at angles of 43.6 and 38.7 , respectively, find the magnitude of each of the two applied forces, to the nearest tenth of
a pound.
Example Regents Questions
1.
Two forces of 25 newtons and 85 newtons acting on a body form an angle of 55 . Find the magnitude of the
resultant force, to the nearest hundredth of a newton. Find the measure, to the nearest degree, of the angle
formed between the resultant and the larger force.
2.
Two applied forces produce a resultant force of 18.6 pounds. The smaller force measures 15.8 pounds, and the
larger force is 24.3 pounds. Find the measure of the angle between the two forces, to the nearest tenth of a
degree.