Download Solving oblique triangles

Solving Oblique Triangles
Revised - Spring 2016
The table below describes how to “solve” oblique triangles. Solving a triangle means finding all angles and
lengths of sides.
C
The angle A is opposite side a.
The angle B is opposite side b.
The angle C is opposite side c.
b
a
A
c
B
You will need the following:
The Law of Sines:
Or, equivalently:
The sum of angles:
sin A sin B sin C
=
=
a
b
c
a
b
c
=
=
sin A sin B sin C
The Law of Cosines:
a 2 = b 2 + c 2 − 2bc cos A
b 2 = c 2 + a 2 − 2ca cos B
c 2 = a 2 + b 2 − 2ab cos C
A+B+C=180
Notes:
1. There can be at most one obtuse angle in a triangle.
2. The longest side is across from the largest angle.
Case #
0
1
Know
3 angles
2 angles, 1 side
1 angle, 2 sides
The angle is across
from one of the sides
2
1 angle, 2 sides
3
The angle is between
the two sides
3 sides
4
Prof. Townsend
Problem solution
No unique solution – all similar triangles work.
i. First find the third angle using A+B+C=180
ii.Then use the Law of Sines twice to find the other two sides.
i. Given x, y, and X, use the Law of Sines to find the angle Y across
from side y. Let h = ysin ( X ) . There are four possible results:
a) x<h
No solution, x is too short.
b) x=h
The triangle is a right triangle.
c) y>x>h There are two solutions to the arcsine, Y and 180-Y, so
there are two possible triangles. Find both.
d) x>y>h There is only one angle Y. 180-Y is negative.
ii. Once two angles are known, find the third from A+B+C=180.
iii. Use the Law of Sines to find the third side.
i. Use the Law of Cosines to find the third side, the one opposite the
given angle.
ii. Find the unknown angle across from the smaller side by the Law
of Sines.
iii. Since the only possible obtuse angle is across from the longest
side, find the largest angle using A+B+C=180 after finding the
other, smaller unknown angle using the Law of Sines.
Find the angle opposite the largest side using the Law of Cosines.
Note that the arccosine returns angles between 0 and 180 so the
answer is unique. Then use the Law of Sines followed by
A+B+C=180 to find the other two angles. Note that the only
possible obtuse angle is across from the longest side, so the
remaining two angles are acute.
7/11/2016
How to identify which case you have by just looking at the parameters you are given.
Case #
0
1
Know
Given Information
3 angles
A, B, C – all capitals
2 angles, 1 side
A, B, a
A, B, b
A, B, c
or
or
or
A, a, C
A, b, C
A, c, C
or
or
or
a, B, C
b, B, C
c, B, C
or
or
Two upper case letters and one lower case letter.
1 angle, 2 sides
2
The angle is across
from one of the sides.
A, a, b
A, a, c
or
or
b, B, a
b, B, c
or
or
a, c, C
b, c, C
or
Two lower case letters and one upper case. One letter has both
lower and upper case present.
1 angle, 2 sides
A, b, c
or
a, B, c
or
a, b, C
3
The angle is between
the two sides.
Two lower case letters and one upper case. All three letters are
present.
4
3 sides
a, b, c - all lower case
The following problems are from Washington on page 288 but in a different order. Identify the case number.
If it is case 2, identify which of the four subcases it is by calculating h as shown above.
Problem 1. A = 67.16°, B = 96.84°, c = 532.9 2. a = 7.86, b = 2.45, C = 2.5° 3. b = 14.5, c = 13.0, C = 56.6° 4. A = 48.0°, B = 68.0°, a = 145 5. a = 186, B = 130.0°, c = 106 6. A = 77.06°, a = 12.07, c = 5.104 7. A = 132.0°, b = 0.750, C = 32.0° 8. a = 22.8, B = 33.5°, C = 125.3° Prof. Townsend
Case Problem 9. b = 750, c = 1100, A = 56° 10. B = 40.6°, b = 7.00, c = 18.0 11. A = 17.85°, B = 154.16°, c = 7863 12. b = 7607, c = 4053, B = 110.09° 13. A = 71.0°, B = 48.5°, c = 8.42 14. a = 1.985, b = 4.189, c = 3.652 15. a = 0.208, c = 0.697, B = 165.4° 16. A = 43.12°, a = 7.893, b = 4.113 Case 7/11/2016