|
Non-right Triangles
Introduction
The following work with all triangles. They are especially useful when
working non-right triangles.
| Law of Sines |
a
b
Sin A Sin B
--------- = -------- or --------- =
-----------
Sin A Sin
B
a
b
The Law of Sines can be used if you know an angle and the side
opposite the angle.
|
| Law of Cosines |
c2 = a2
+ b2 - 2ab Cos C
a2 = c2 + b2
- 2bc Cos A
b2 = a2 + c2
- 2ac Cos B
The Law of Cosines can be used if you know
a. Two sides and the angle between the two sides.
b. All three sides. |
The Law of Sines is generally quicker to use than the Law of
Cosines. Use the Law of Sines when possible.
A + B + C = 180o If you know any
two angles, you can solve for the third angle.
In many problems you will start with one law and finish the problem with
the other law. You cannot use the definitions for Sine, Cosine, and
Tangent with non-right triangles.
Table of Contents
Non-right
Triangle Problems
| 1. |
A is 35.6o, b is 7.55 cm, and B
is 120.1o a =
?, C = ?, and c = ?
|
| 2. |
c is 31.3 cm, b is 52.4 cm, and B is 105.0o
C = ?, A = ?, and a = ?
|
| 3. |
a is 8.33 cm, b is 16.4 cm, and c is 19.4
cm A = ?, B = ?, and C = ?
|
| 4. |
a is 9.22 cm, b is 13.9 cm, and C is 66.3o
A = ?, c = ?, and B = ? |
Table of Contents
Non-Right
Triangle Answers
| 1. |
A is 35.6o, b is 7.55
cm, and B is 120.1o a
= ?, C = ?, and c = ?
|
|
a
b
b Sin A
-------- = -------- => a = -------------
Sin A Sin
B
Sin B
7.55 cm x
Sin 35.6o
a = --------------------------- = 5.08 cm
Sin
120.1o
C = 180o - 35.6o
- 120.1o = 24.3o
c
b
b Sin C
-------- = -------- => c = -------------
Sin C Sin
B
Sin B
7.55 cm x
Sin 24.3o
c = --------------------------- = 3.59 cm
Sin
120.1o |
There
is a known side-angle combination. (b and B).
Solve Law of Sines for a
A + B + C = 180o
Solve for C
The Law of Sines can now be solved for c. |
|
|
|
| 2. |
c is 31.3 cm, b is 52.4 cm, and
B is 105.0o C = ?, A = ?, and a = ?
|
|
Sin
C Sin
B
c Sin B
-------- = ---------- => C = Sin -1
( -------------- )
c
b
b
31.3 cm x Sin 105.0o
C = Sin -1 ( ----------------------------- ) = 35.2o
52.4 cm
C = 180o - 105.0o - 35.2o
= 39.8o
c
b
b Sin C
-------- = -------- => c = -------------
Sin C Sin
B
Sin B
52.4 cm x
Sin 39.8o
c = --------------------------- = 34.7 cm
Sin
105.0o |
There
is a known side-angle combination. (b and B).
Solve Law of Sines for C.
A + B + C = 180o
Solve for C
The Law of Sines can now be solved for c.
|
|
|
|
| 3. |
a is 8.33 cm, b is 16.4 cm, and
c is 19.4 cm A = ?, B = ?, and C = ?
|
|
a2 + b2 - c2
c2 = a2 + b2 - 2ab Cos
C => C = Cos -1 ( ---------------------- )
2ab
(8.33 cm)2 + (16.4 cm)2 - (19.4 cm)2
C = Cos -1 (
----------------------------------------------------- ) = 98.0o
2 x 8.33 cm x 16.4 cm
Sin B Sin
C
b Sin C
------- = --------- => B = Sin -1 ( ------------
)
b
c
c
16.4 cm x Sin 98.0o
B = Sin -1 ( -------------------------- ) = 56.8o
19.4 cm
A = 180o - 98.0o
- 56.8o = 25.2o |
All
three sides are known.
Solve the Law of Cosines for C
Solve Law of Sines for B
Solve A + B + C = 180o for
A |
|
|
|
| 4. |
a is 9.22 cm, b is 13.9 cm, and
C is 66.3o A = ?, c =
?, and B = ?
|
|
c2
= a2 + b2 - 2ab Cos C
=> c = ( a2 + b2 - 2ab Cos C )
1/2
c = ( (9.22 cm)2 +
(13.9 cm)2 - 2 x 9.22 cm x 13.9 cm x Cos 66.3o)
1/2
c = 13.2 cm
Sin A Sin
C
a Sin C
-------- = ---------- => A = Sin -1 ( ------------- )
a
c
c
9.22 cm x Sin 66.3o
A = Sin -1 ( ---------------------------- ) = 39.8o
13.2 cm
B = 180o - 66.3o - 39.8o
= 73.9o |
Two
sides and the angle between the two sides is known.
Solve Law of Cosines for c
Solve Law of Sines for A
180o = A + B + C Solve
for B |
Table of Contents
|
