Plane Trigonometry - Lecture 16 Section 3.2: the Law of Cosines

Plane Trigonometry - Lecture 16 Section 3.2: The Law of Cosines

Summary: http://www.math.ksu.edu/˜gerald/math150/sum16.pdf Course page: http://www.math.ksu.edu/˜gerald/math150/

Gerald Hoehn

April 1, 2019 Law of cosines

Theorem Let ∆ABC any triangle, then

c2 = a2 + b2 − 2ab cos γ b2 = a2 + c2 − 2ac cos β a2 = b2 + c2 − 2bc cos α

We may reformulate the statement also in word form. Theorem In any triangle, the square of the length of a side equals the sum of the squares of the length of the other two sides minus twice the product of the length of the other two sides and the cosine of the angle between them. Solving Triangles

For solving triangles ∆ABC one needs at least three of the six quantities a, b, and c and α, β, γ. One distinguishes six essential diﬀerent cases forming three classes:

I AAA case: Three angles given. I AAS case: Two angles and a side opposite one of them given. I ASA case: Two angles and the side between them given. I SSA case: Two sides and an angle opposite one of them given. I SAS case: Two sides and the angle between them given. I SSS case: Three sides given. The case AAA cannot be solved. The cases AAS, ASA and SSA are solved by using the law of sines. The cases SAS, SSS are solved by using the law of cosines. Solving Triangles: the SAS case

For the SAS case a unique solution always exists. Three steps: 1. Use the law of cosines to determine the length of the third side opposite to the given angle. 2. Use the law of sines to determine the missing angle opposite to the shorter of the two given sides. 3. Use the formula α + β + γ = 180◦ to determine the third angle.

Applying the law of sines to the missing angle opposite to the shorter of the two given sides guarantees that the angle is acute and the need of discussing two solutions. Solving Triangles: the SSS case For the SSS case a unique solution exists if the sum of the lengths of two sides is larger than the remaining. Otherwise no solution exists. Three steps: 1. The law of cosines is applied to determine the angle opposite of the longest side. 2. The second angle is obtained either by the law of sines or the law of cosines. 3. The formula α + β + γ = 180◦ is used to determine the third angle.

Applying the law of cosines to the angle opposite to the longest of the three given sides guarantees that the remaining two angle are acute so that the inverse sine function gives the correct solution. If using the law of cosines, this problem does not arise since by using the arccosine one obtains a unique angle between 0◦ and 180◦. The area of a triangle The law of the cosines allows to compute the area of a triangle from the given data in the SAS and SSS cases.

Theorem The area A of a triangle ∆ABC equals SAS 1 A = bc sin α 2 1 A = ac sin β 2 1 A = ab sin γ 2 SSS A = ps (s − a)(s − b)(s − c) 1 where s is half of the perimeter: s = 2 (a + b + c).

For the other cases one ﬁrst computes additional data.