Section 7.3 - The Law of Sines and the Law of Cosines
Sometimes you will need to solve a triangle that is not a right triangle. This type of
triangle is called an oblique triangle. To solve an oblique triangle you will not be able to
use right triangle trigonometry. Instead, you will use the Law of Sines and/or the Law of
Cosines.
You will typically be given three parts of the triangle and you will be asked to find the
other three. The approach you will take to the problem will depend on the information
that is given.
If you are given SSS (the lengths of all three sides) or SAS (the lengths of two sides and
the measure of the included angle), you will use the Law of Cosines to solve the triangle.
If you are given SAA (the measures of two angles and one side) or SSA (the measures of
two sides and the measure of an angle that is not the included angle), you will use the
Law of Sines to solve the triangle.
Recall from your geometry course that SSA does not necessarily determine a triangle.
We will need to take special care when this is the given information.
Please read this before class!
Here are some facts about solving triangles that may be helpful in this section:
If you are given SSS, SAS or SAA, the information determines a unique triangle.
If you are given SSA, the information given may determine 0, 1 or 2 triangles. This is
called the “ambiguous case” of the law of sines. If this is the information you are given,
you will have some additional work to do.
Since you will have three pieces of information to find when solving a triangle, it is
possible for you to use both the Law of Sines and the Law of Cosines in the same
problem.
When drawing a triangle, the measure of the largest angle is opposite the longest side; the
measure of the middle-sized angle is opposite the middle-sized side; and the measure of
the smallest angle is opposite the shortest side.
Suppose a, b and c are suggested to be the lengths of the three sides of a triangle.
Suppose that c is the biggest of the three measures. In order for a, b and c to form a
triangle, this inequality must be true: a + b > c . So, the sum of the two smaller sides
must be greater than the third side.
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An obtuse triangle is a triangle which has one angle that is greater than 90°. An acute
triangle is a triangle in which all three angles measure less than 90°.
If you are given the lengths of the three sides of a triangle, where c > a and c > b, you
can determine if the triangle is obtuse or acute using the following:
If a 2  b 2  c 2 , the triangle is an acute triangle.
If a 2  b 2  c 2 , the triangle is an obtuse triangle.
Your first task will be to analyze the given information to determine which formula to
use. You should sketch the triangle and label it with the given information to help you
see what you need to find. If you have a choice, it is usually best to find the largest angle
first.
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THE LAW OF SINES AND THE LAW OF COSINES
Here’s the Law of Sines. In any triangle ABC,
C
b
A
a
c
B
sin A sin B sin C


.
a
b
c
USED FOR SAA, SSA cases!
SAA: One side and two angles are given
SSA: Two sides and an angle opposite to one of those sides are given
Example 1: Find x.
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Here’s the Law of Cosines. In any triangle ABC,
C
b
A
a
c
B
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
USED FOR SAS , SSS cases!
SAS: Two sides and the included angle are given
SSS: Three sides are given
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Example 2: In ABC , a = 5, b = 8, and c = 11. Find the measures of the three angles to
the nearest tenth of a degree.
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Example 3: In XYZ , X  26, Z  78 and y  18. Solve the triangle. Give exact
answers.
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Example 4: In ABC , A  50, b  9 and a  6. Solve the triangle and round all
answers to the nearest hundredth.
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Example 5: Two sailboats leave the same dock together traveling on courses that have an
angle of 135between them. If each sailboat has traveled 3 miles, how far apart are the
sailboats from each other?
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Example 6: In ABC , B  60 , a = 17 and c = 12. Find the length of AC.
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Note: SSA case is called the ambiguous case of the law of sines. There may be two
solutions, one solution, or no solutions. You should throw out the results that don’t make
sense. That is, if sin A 1 or the angles add up to more than 1800.
SSA Case (Two sides and an angle opposite to those sides)
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Example 7: In PQR, P  112, p  5 and q  7. How many possible triangles are
there? Solve the triangle. Round the answers to three decimal places.
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Example 8: In XYZ , Y  22, y  7, x  5 . How many possible triangles are there?
Solve the triangle and round all answers to the nearest hundredth.
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MATH 1330 Review for Test -4
Time: 50 minutes
Number of questions: 11 Multiple Choice Questions (total: 100 pts)
What is covered: 5.1, 4.4, Chapters 6 and 7.
Do not forget to reserve a seat for Test – 4!
Take practice Test – 4! 10% of your best score will be added to your test grade.
Remember the make-up policy: No make-ups!
The following formulas will be provided. It is your responsibility to locate the formula sheet
before you start your test. If you can’t find it, ask proctors for help.
Handy Formulas
sin (s + t) = sin s cos t + cos s sin t
sin (s − t) = sin s cos t − cos s sin t
cos (s + t) = cos s cos t − sin s sin t
cos (s − t) = cos s cos t + sin s sin t
tan (s + t) =
tan ( s − t) =
tan s + tan t
1 − tan s tan t
tan s − tan t
1 + tan s tan t
cos(2t) = cos 2 t − sin 2 t
sin(2t) = 2sin t cos t
sin
1 − cos s
s
=±
2
2
cos
1 + cos s
s
=±
2
2
tan
sin s
s
=
2 1 + cos s
1
cos(− x)
cot(− x)
1) Simplify:
−
2) Simplify:
cos 2 ( x) +
tan( x) − cot( x)
tan( x) + cot( x)
2
3) Simplify:
tan( x) −
cos( x)
1 − sin( x)
3
4) Given sin( x) =
1
2
, 90 0 < x < 180 0 , and sin( y ) = − , 180 0 < y < 270 0 ,
4
5
Find:
a) sin( x − y )
b) sin( x + y )
c) cos( x − y )
d) cos( x + y )
4
5) Given tan( x) = −
1
, 90 0 < x < 180 0 ,
5
a) sin(2 x)
b) cos(2 x)
5
6) Given sin( x) =
3
 x
, where x is an acute angle, find sin   .
5
2
7) Find the following using the sum or difference formulas:
a) cos(750 )
b) sin(105 0 )
6
8) Solve the following equation over the interval [0,2π ) : 2 sin 2 ( x) + 9 sin( x) + 7 = 0
7
9) Find all solutions of the equation cos(4 x) =
1
over the interval [0,2π ) .
2
10) Find the area of triangle ABC if ∠B = 30° , c = 10 and a = 12.
8
11) ABC is a triangle with AB = 10, BC = 13, and AC = 7. Find cos(A).
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12) The angle of elevation from a point that is 120 ft away from a building to the top of the building
is 250. Find the height of the building.
10
13) Two boats leave the dock at the same time and they travel with an angle of 1200 between them.
What is the distance between them after they each travel 4 miles?
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14) Given a triangle ABC, A = 450, B = 300, BC = 60 cm, find AC.
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15- a) Given a triangle ABC with AB = 6 cm and BC = 4, angle A measures 1200.
How many choices are there for the measure of angle C?
b) Given a triangle ABC with AB = 6 cm and BC = 4, angle A measures 300.
How many choices are there for the measure of angle C?
c) In triangle ABC with AB = 6, BC = 7, the measure of angle A is 30o. How many choices are
there for the measure of angle C?
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