Download Chapter 14 Formulas for Non-right triangles Area of a triangle: need

Chapter 14
Formulas for Non-right triangles
Area of a triangle: need two sides and the included angle
1
k = absinC
2
where a and b are any two sides and C is the angle between
them.
1
2
1
2
**This formula can also be written as: k = acSinB or k = cbSinA **
1
2
Ex. Find the area of ΔABC, given a = 16,b = 12, and sinC = .
Ex. If the area of ΔDEF is 101 square centimeters, d = 16 and e = 14, find the
measure of F to the nearest ten minutes or tenth of a degree.
Ex. Find the area of the given triangle, to the nearest hundredth of a square
meter.
Ex. In parallelogram HOPE, HO = 16.7 cm, OP = 20.1 cm, and m HOP = 108.3
a. Find the length of HP to the nearest tenth of a centimeter.
b. Find the area of HOPE to the nearest hundredth of a square centimeter.
Ex. In isosceles triangle CAT, CA=AT, m C = 72, and c= 6.3 inches. Find the
area of CAT to the nearest hundredth.
If we are given two sides in a right triangle, we would have no problem
finding the missing side: we would use the Pythagorean theorem. However, if we
do not have a right triangle, we could not use the Pythagorean thm. We have
some new trig laws that we can use now.
1. Law of Cosines- used when we do not have a right triangle and we are given
either S.A.S (two sides and the included angle) or S.S.S ( all three sides).
The law of cosine may be written in different forms, depending on the sides
and angles given. (the variables will change within the formula)
a 2 = b 2 + c 2 − 2bcCosA
b 2 = a 2 + c 2 − 2acCosB
c 2 = a 2 + b 2 − 2abCosC
ex. In triangle HAT, a = 6.4, t = 10.2, and m∠H = 87. Find the length of side h to the
nearest tenth and then find the area of HAT.
Ex. In isosceles triangle QRS, q = s = 4.7centimeters. If cos R = 0.1908, find the length of
side r to the nearest hundredth of a centimeter.
Ex. Mr. Smith wants to make his backyard garden unusual, so he decides to
design it in the shape of an obtuse triangle with white flowers acting as the border
on the two shorter sides of the triangle and purple flowers along the length of the
longer side. The two shorter sides measure 8 feet and 9 feet, including an angle
of 105 degrees.
a. How long, to the nearest tenth of a foot, must the third side of the triangle
be?
b. If Mr. Smith needs 4 purple flower plants for each foot of the border, how
many plants will he need?
Ex. 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 of a degree, what is the largest angle he needs to create using the
lead caning?
Ex. Three sides of a triangle measure 5, 8, and 12. Is the triangle isosceles, right,
acute or obtuse?
Ex. Dimitri and Anna are in charge of setting the route for the Daffy Drivers’ Bike
Race at the County Fair. This is an event for the 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 is 1.7 miles, the distance from
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.
2. Law of Sines: This is used when we don’t have S.A.S or S.S.S.
A proportion is set up for this formula:
a
b
c
=
=
SinA sin B SinC
or
SinA SinB SinC
=
=
a
b
c
** sometimes you will be given the angle and sometimes you will be given
the sine of the angle, be careful**
ex. In triangle ANT, m∠ANT = 66.5, m∠NTA = 47, and NT = 12.4 inches. Find the length of
NA to the nearest tenth of an inch.
Ex. In triangle CTH, m∠T = 107.3, m∠H = 34.5, and CH = 17.2 cm. Find the length of CT
to the nearest ten thousandth.
Ex. Given ΔJKL in which JK = 14.3, KL = 5.8, and m∠LJK = 17, find the measure of
∠KLJ to the nearest tenth of a degree or nearest ten minutes.
(This example will introduce the ambiguous case)
Ex. If m = 7 , n = 10, and m∠M = 85, how many different triangles MNP can be
drawn?
Forces and Vectors:
Applied forces
Resultant
Magnitude
Ex. Two forces of 28 pounds and 41 pounds act on a body so that the angle
between them measures 72 degrees. Find, to the nearest tenth of a pound, the
magnitude of the resultant the forces produce.
Draw a diagram
Label all the givens
Fill in the top angle
Use the top or bottom triangle
Use the Law of Sines or the Law of Cosines to solve the problem
Ex. 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.
Ex. 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 degrees and 38.7
degrees with resultant, find the magnitude of each of the two applied forces to the
nearest tenth of a pound.
Mixed Trigonometric Applications:
Ex. An isosceles triangle has base angles of 53.4 degrees and a base equal to
14.7 inches. Find, to the nearest tenth of an inch, the length of the equal sides of
the triangle.
Ex. Jamie is working with stained glass and she needs to cut an obtuse triangle
with sides of 11.6 cm and 8.4 cm to fit into her design. If the area of the triangle
must be 37.2 square cm, to the nearest tenth of a degree, along what obtuse
angle must she cut the glass?
Ex. Two forces act on a body forming a resultant force of 46 pounds. If the angle
between the resultant and the smaller force of 19.8 pounds is 54.9 degrees, what
is the magnitude of the larger force, to the nearest tenth of a pound?
Ex. Katie is out with her parents at the fair when she sees a large balloon with her
name on it. Her dad tells her the angle of elevation from where she is standing to
the foot of the balloon is 32 degrees but Katie is in too much of a hurry to get
closer to the balloon to listen. She runs 120 feet toward the area where the
balloon is hovering before her mom, a math teacher, catches up with her and says
that the angle of elevation from where she is now to the foot of the balloon is 54
degrees. But Katie wants to know only one thing. “ I want to go up there. How
high is it?” she asks. Answer Katie’s question to the nearest tenth of a foot.