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```UNIT 36
FUNCTIONS OF ANY ANGLE, OBLIQUE
TRIANGLES
OBLIQUE TRIANGLES
An oblique triangle is a triangle that does not
have a right angle
 An oblique triangle may be either acute or obtuse
 In an acute triangle, each of the three angles is
acute or less than 90°
 In an obtuse triangle, one of the angles is obtuse
or greater than 90°

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LAW OF SINES
 In
any triangle the sides are proportional to the
sines of their opposite angles or, stated as a
formula:
a

sin A
b

sin B
c
sin C
C
b
A
a
B
c
Pay attention to how A and a are across from each other!
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LAW OF SINES

The Law of Sines is used to solve the following
two kinds of oblique triangle problems:
1.
Problems where any two angles and any side of an
oblique triangle are known
2.
Problems where any two sides and an angle
opposite one of the given sides of an oblique
triangle are known
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SOLVING TRIANGLES GIVEN TWO
ANGLES AND A SIDE

Solve the oblique triangle shown below for side x:
– Set up the proportion using the Law
of Sines:
x
6 . 7 cm
49°
47°
sin 49


x
sin 47

– Solve the proportion:
6.7(sin 47°) = (sin 49°)x
x = 6.49 cm Ans
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AMBIGUOUS CASE
It is possible to have two different triangles with
the same two sides and the same angle opposite
one of the given sides. A situation of this kind is
called an ambiguous case
 The only conditions under which a problem can
have two solutions is when the given angle is
acute and the given side opposite the given angle
is smaller than the other given side
 Since most problems do not involve two solutions,
this situation will not be discussed in any more
detail

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SOLVING TRIANGLES GIVEN TWO SIDES
AND AN OPPOSITE ANGLE

Solve for angle C in an oblique triangle given that
A is 101, side a is 25 inches, and side c is 14
inches
– First, draw the triangle and label the given parts
– Now set up the proportion:
A
14"
B
101°
25"
b
25 "
sin 101


14 "
sin C
C – Now, solving for angle C yields:
C = 33.3 Ans
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LAW OF COSINES

In any triangle, the square of any side is equal to
the sum of the squares of the other two sides minus
twice the product of these two sides multiplied by
the cosine of their included angle
a2 = b2 + c2 – 2bc(cos A)
b2 = a2 + c2 – 2ac(cos B)
c2 = a2 + b2 – 2ab(cos C)

The Law of Cosines is used to solve oblique triangle
problems where two sides and the included angle
are known or when all three sides are known
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SOLVING PROBLEMS GIVEN TWO SIDES
AND THE INCLUDED ANGLE

Solve for side c in the oblique triangle shown below
using the Law of Cosines:
42 mm
– Use the appropriate law:
c
c  a  b  2 ab (cos C )
2
2
2
– Now, substitute the given values:
49 mm
c  42  49  2 ( 42 )( 49 )(cos 87 )
2
2

2
– Solving for c yields:
c = 62.85 mm Ans
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LAW OF COSINE PROBLEMS GIVEN THREE
SIDES OF A TRIANGLE

Find angle A in the triangle shown below:
A
– Use the appropriate law in order
to find the largest angle first:
a  b  c  2 bc (cos A ) or
2
29"
2
cos A 
2
29
2
 15
2
 16
2
(  2 )( 15 )( 16 ) 
– Solving for angle A yields:
A = 138.6 Ans
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PRACTICE PROBLEMS

Solve problems 1–3 using the Law of Sines.
Round all angles to the nearest tenth of a degree
and all sides to two decimal places.
1.
Find side x in the figure below:
89°
x
48°
29 cm
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PRACTICE PROBLEMS (CONT)
2.
Given an oblique triangle with C = 101, side c =
35 inches, and side a = 10 inches, solve for A, B,
and side b.
3.
Given an oblique triangle with A = 115, C =
23, and side a = 47 cm, find side c, side b, and
angle B.
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PRACTICE PROBLEMS (CONT)

Solve problem 4 using the Law of Cosines.
Round all angles to the nearest tenth of a
degree and all sides to two decimal places.
4.
Completely solve the triangle shown below:
a
C
B
4.9"
98°
4.7"
A
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PRACTICE PROBLEMS (CONT)

Solve problem 5 using the Law of Cosines.
Round all angles to the nearest tenth of a
degree and all sides to two decimal places.
5.
6.
Determine all the angles of an oblique triangle
given that side a is 3.5 feet, side b is 2.7 feet, and
side c is 1.2 feet.
A triangular patio has sides of 12 feet, 8 feet, and
7.5 feet. Determine its largest angle.
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PROBLEM ANSWER KEY
1.
2.
3.
4.
5.
6.
x = 21.55 cm
A = 16.3, B = 62.7, and b = 31.68 inches
B = 42, c = 20.26 cm, and b = 34.70 cm
B = 42, C = 40, and a = 7.25 inches
A = 122.9, B = 40.4, and C = 16.7
101.4°
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