Math 120 - Cooley Trigonometry OCC
Section 4.2 – The Law of Sines
Law of Sines
For a triangle with sides a, b, c, and opposite angles A, B, C, respectively,
If a triangle is not a right triangle then it is called an oblique triangle.
There are two types of oblique triangles:
One with three acute angles or one with exactly one obtuse angle and two acute angles.
To solve an oblique triangle means to find the lengths of all three sides and the measures of all three angles. To do this, we shall need to know the length of one side along with (i) two angles; (ii) one angle and one other side; or (iii) the other two sides. There are four possibilities to consider.
CASE #1: One side and two angles are known (ASA or SAA)……………………… {USE LAW OF SINES}
CASE #2: Two sides and the angle opposite one of them are known (SSA)………… {USE LAW OF SINES}
CASE #3: Two sides and the included angle are known (SAS)…………...... {USE LAW OF COSINES}
CASE #4: Three sides are known (SSS)………………………………………….. {USE LAW OF COSINES}
(Note: There is no such postulate or theorem AAA, because this would result in a family of similar triangles).
The figure below illustrates the 4 cases. (Any S and/or A labeled represents that those measurement are known).
CASE #1: ASA CASE #1: SAA CASE #2: SSA CASE #3: SAS CASE #4: SSS
CASE #2 – SSA – AMBIGUOUS CASE
SSA is known as the ambiguous case, because the known information may result in one triangle, two triangles, or no triangle at all.
Suppose that sides a and b along with angle A is given or known. Then here are the possibilities that can occur:
NO TRIANGLE – (where )
ONE RIGHT TRIANGLE – (where )
TWO TRIANGLES – (where )
ONE TRIANGLE – (where )
J Exercises: Solve each triangle.
1) A = 50°, C = 20°, a = 3
2) B = 20°, C = 70°, a = 1
J Exercises: Two sides and an angle are given. Determine whether the given information results in one
triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.
3) b = 4, c = 3, B = 40°
4) b = 4, c = 5, B = 61°
5) a = 2, c = 1, A = 120°
6) b = 2, c = 3, B = 40°
J Exercises:
7) A loading ramp 10 feet long that makes an angle of 18° with the horizontal is to be replaced by one that
makes an angle of 12° with the horizontal. How long is the new ramp?
8) In attempting to fly from city P to city Q, an aircraft followed a course that was 10° in error, as indicated in
the figure. After flying a distance of 50 miles, the pilot corrected the course by turning at a point R and
flying 70 miles farther. If the constant speed of the aircraft was 250 miles per hour, how much time was lost
due to the error?
- 4 -