Unit 2 - Objective 5 - Oblique Triangles & Laws of Sines
Quick Review

Triangles that do not have a right angle are called oblique triangles. The trigonometric methods for solving right triangles do not work with oblique triangles. There are two methods that are usually used with oblique triangles: law of sines and law of cosines (objective 6).

The law of sines


The law of sines are used to solve the following two cases:

  1. When the measure of two sides and the angle opposite one of them is known, and
  2. When the measure of two angles and one side is known.



Example 1

Given: A = 33°, a = 9.4 and c = 14.3
Solve the triangle.

Solution

We are given:
sidesangles
a = 9.4A = 33°
b = ?B = ?
c = 14.3C = ?

Using the law of sines:


B = 180° - 33° - 55.95° = 91.05°

We can now find the length of the third side.


Now we have:
sidesangles
a = 9.4A = 33°
b = 17.26B = 91.05°
c = 14.3C = 55.96°




Example 2

Given: A = 82.17°, B = 64.43° and c = 9.12
Solve the triangle.

Solution

We are given:
sidesangles
a = ?A = 82.17°
b = ?B = 64.43°
c = 9.12C = ?

We can find angle C:

C = 180° - 82.17° - 64.43° = 33.40°

Now use the law of sines:


Now find side b using the law of sines:


Now we have:
sidesangles
a = 16.41A = 82.17°
b = 14.94B = 64.43°
c = 9.12C = 33.40°




Example 3

Given: a = 20, b = 24, and A = 55.4°
Solve the triangle.

Solution

We are given:
sidesangles
a = 20A = 55.4°
b = 24B = ?
c = ?C = ?


Since sin theta is positive in the 1st and 2nd quadrants,
then B could also be 180° - 81.03° = 98.97°.

Now we have:
sidesangles
a = 20A = 55.4°
b = 24B = 81.03°
c = ?C = 43.57°

sidesangles
a = 20A = 55.4°
b = 24B = 98.97°
c = ?C = 25.63°

C = 180° - 55.4° - 81.03°
   = 43.57°

C = 180° - 55.4° - 98.97°
   = 25.63°

We now need to use the law of sines with C = 43.57° and C = 25.63°:


C sin 55.40° = 20 sin 43.57°
C sin 55.40° = 20 sin 25.63°

Finally we have:
sidesangles
a = 20A = 55.4°
b = 24B = 81.03°
c = 16.75C = 43.57°

sidesangles
a = 20A = 55.4°
b = 24B = 98.97°
c = 10.51C = 25.63°

Note:

If sin theta is larger than 1, then the triangle is impossible.



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