7.3 The Law of Cosines

Triangle Side Length Restriction

In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Example

Is it possible to construct a triangle with sides of length 2,6, and 9?

Law of Cosines

In any triangle ABC, with sides a, b, and c,

Use the Law of Cosines to solve SAS and SSS Triangles (Cases 3 & 4)

Example

Two boats leave a harbor at the same time, traveling on courses that make an angle of between them. When the slower boat has traveled 62.5 km, the faster one has traveled 79.4 km. At that time what is the distance between the boats?

Example

Solve triangle ABC if

Note: Once the law of cosines has been used, the ambiguous case of the law of sines does not occur.

Example

Solve triangle ABC if a = 25.4 cm, b = 42.8 cm, and c = 59.3 cm.

Find angle C for the truss shown in the figure below.

Summary of four possible cases when solving an oblique triangle.

Oblique Triangle	Suggest Procedure for Solving
Case I: One side and two angles are known. (SAA or ASA)	Step 1: Find the remaining angle using sum of angles of triangle. Step 2: Find the remaining sides using the law of sines.
Case 2: Two sides and one angle (not included between the two sides) are known. (SSA)	This is the ambiguous case!! Step 1: Find an angle using the law of sines Step 2: Find the remaining angle using sum of the angles of triangle. Step 3: Find the remaining side using the law of sines. (If a 2^nd triangle exists, repeat)
Case 3: Two sides and the included angle are known. (SAS)	Step 1: Find the third side using the law of cosines. Step 2: Find the smaller of the two remaining angles using the law of sines. Step 3: Find the remaining angle using the sum of angles of triangle.
Case4: Three sides are known (SSS)	Step 1 Find the largest angle using the law of cosines. Step 2 Find either remaining angle using the law of sines. Step 3: Find the remaining angle using the sum of angles of triangle.

Outline the steps to take to solve. Do not actually solve.

1.

2.

3.

4.

5.

Heron’s Formula for the Area of a Triangle

If a triangle has sides of lengths a, b, and c, with semi perimeter

then the area of the triangle is

Example

The distance “as the crow flies” from Chicago to St. Louis is 262 m, from St. Louis to New Orleans is 599 miles, and from New Orleans to Chicago is 834 miles. What is the area of the triangular region having these three cities as vertices?