6.1 Law of Sines
To solve an oblique triangle, at least one side and the measures of any two other parts of the triangle must be known Two sides, two angles, or one angle and one side. This breaks down into the following 4 cases
The first to cases can be solved using the Law of Sines, whereas the last two can be solved using the Law of Cosines.
- Two angles and any side (AAS or ASA)
- Two sides and an angle opposite one of them (SSA)
- Three sides (SSS)
- Two sides and their included angle (SAS)
The first to cases can be solved using the Law of Sines, whereas the last two can be solved using the Law of Cosines.
Given Two Angles and One Side AAS
Given Two Angles and One Side ASA
A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14° with the horizontal. The flagpole casts a 16 meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20°.
(a) Draw a triangle that represents the problem. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation involving the unknown quantity. (c) Find the height of the flagpole. |
The Ambiguous Case (SSA)
If two sides and one opposite angle are given, then three possible situations can occur
no such triangle exists
one such triangle exists
two distinct triangles satisfy the conditions.
If two sides and one opposite angle are given, then three possible situations can occur
no such triangle exists
one such triangle exists
two distinct triangles satisfy the conditions.
Single Solution Case SSA
No Solution Case SSA
Two Solution Case SSA
Area of an Oblique Triangle
The procedure used to prove the Law of Sines leads to a simple formula for the area of an oblique triangle. Each triangle has a height of h = b sin A.
To see this when A is obtuse, substitute the reference angle 180° - A for A. The height of the triangle is given by
h= b sin (sin 180° cos A - cos 180° sin A)
= b [(0)(cos A - (-1))(sin A)]
=b Sin A
So the area of each triangle is given by
Area = 1/2 (base)(height)
= 1/2 (c)(b sin A)
= 1/2 bc sin A
The formulas can be developed
Area = 1/2 ab sin C = 1/2 ac sin B
The procedure used to prove the Law of Sines leads to a simple formula for the area of an oblique triangle. Each triangle has a height of h = b sin A.
To see this when A is obtuse, substitute the reference angle 180° - A for A. The height of the triangle is given by
h= b sin (sin 180° cos A - cos 180° sin A)
= b [(0)(cos A - (-1))(sin A)]
=b Sin A
So the area of each triangle is given by
Area = 1/2 (base)(height)
= 1/2 (c)(b sin A)
= 1/2 bc sin A
The formulas can be developed
Area = 1/2 ab sin C = 1/2 ac sin B
Finding the Area of an Oblique Triangle
Application of the Law of Sines