Math 170 - Cooley Pre-Calculus OCC
Section 7.1 – The Law of Sines
Theorem: The Law of Sines:
For a triangle with sides a, b, c, and opposite angles , , , 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 we are given an acute angle A and sides a and b. Side a is opposite the angle A, and side b is adjacent to it. Draw an angle of approximate size A in standard position with terminal side of length b. Don’t draw in side a yet. Let h be the distance from C to the initial side of A. Since , we have .Now, we are ready to draw in side a, but there are four possibilities for its location as illustrated below:
NO TRIANGLE – (where )
ONE RIGHT TRIANGLE – (where )
TWO TRIANGLES – (where )
ONE TRIANGLE – (where )
The Area of a Triangle
The area of a triangle is
The Area of a SAS Triangle
The area of a SAS triangle is
J Exercises: Solve each triangle with the given parts.
1) , ,
2) , ,
Determine the number of triangles with the given parts and solve each triangle.
3) , ,
J Exercises:
Determine the number of triangles with the given parts and solve each triangle.
4) , ,
5) , ,
J Exercises: Find the area of each triangle with the given parts.
6) , ,
7) , ,
J Exercises: Find the area of each triangle with the given parts.
7) Joe and Jill set sail from the same point, with Joe sailing in the same direction of S4°E and Jill sailing in
the direction S9°W. After 4 hr, Jill was 2 mi due west of Joe. How far had Jill sailed?
8) The F-106 Delta Dart once held a world record of Mach 2.3. Its sweptback triangular wings have the
dimensions given in the figure. Find the area of one wing in square feet.
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