Comprehensive Study Guide: The Law of Cosines

Instructional Objectives and Educational Framework

The mathematical study of oblique triangles involves understanding the structural relations between side lengths and angle measures. Under the educational framework of the Department of Education (Edukasyon, Republika ng Pilipinas, Bagong Pilipinas) presented by John Cedric A. Jacobo, Teacher III, Mathematics, the primary objective is to illustrate and apply the Law of Cosines. Learners are expected to analyze and demonstrate oblique triangles under two specific given conditions: when two side measures and an included angle are provided, and when all three side measures of a triangle are given.

Review of Oblique Triangles and Law Selection Criteria

Determining whether to apply the Law of Sines or the Law of Cosines depends on identifying the specific combination of given sides and angles in an oblique triangle. There are three specific configurations where the Law of Sines is applicable: Angle-Side-Angle (ASA), Angle-Angle-Side (AAS or SAA), and Side-Side-Angle (SSA or ASS). If an oblique triangle setup does not match one of these three configurations, the Law of Sines cannot be used, requiring the application of the Law of Cosines instead.

The primary cases where the Law of Sines fails and the Law of Cosines must be applied are Side-Side-Side (SSS) and Side-Angle-Side (SAS). Although both SSA (or ASS) and SAS involve two known side lengths and one known angle, they are fundamentally distinct. In an SSA or ASS configuration, the known angle is not included between the two known sides, which allows the Law of Sines to be applied. In an SAS configuration, the known angle must be the included angle—the angle formed precisely at the vertex where the two adjacent known sides intersect.

Evaluating individual given cases yields specific operational decisions for oblique triangles: An Angle-Side-Angle (ASA) configuration can be solved using the Law of Sines. A Side-Side-Side (SSS) configuration cannot be solved using the Law of Sines and must be solved using the Law of Cosines. A Side-Side-Angle (SSA) configuration can be solved using the Law of Sines. A Side-Angle-Side (SAS) configuration cannot be solved using the Law of Sines and must be solved using the Law of Cosines. A Side-Angle-Angle (SAA) configuration can be solved using the Law of Sines.

Conceptual Exploration and Mathematical Formulation of the Law of Cosines

To understand why the Law of Sines fails in an SAS configuration, consider a scenario where Matt and Jack start biking from the same point $A$ . Matt bikes a distance of $4\,km$ while Jack bikes a distance of $15\,km$ . The two distinct paths covered by the bikers form an angle of $112^\circ$ at the starting vertex $A$ . The objective is to determine the straight-line distance between the two riders. This problem gives two side lengths ( $4\,km$ and $15\,km$ ) and their included angle ( $112^\circ$ ), establishing an SAS case. Attempting to apply the Law of Sines to this arrangement results in an equation containing two unknown variables in a single equation, making it impossible to solve directly. Therefore, the Law of Cosines must be utilized.

The Law of Cosines states that the square of any one side of a triangle is equal to the difference between the sum of the squares of the other two sides and double the product of those two sides and the cosine of the angle included between them.

To compute unknown side lengths in a triangle with sides $a$ , $b$ , and $c$ , and opposite interior angles $A$ , $B$ , and $C$ , the side formulas are expressed as: $a^2 = b^2 + c^2 - 2bc \cos(A)$ $b^2 = a^2 + c^2 - 2ac \cos(B)$ $c^2 = a^2 + b^2 - 2ab \cos(C)$

To compute unknown interior angles when side lengths are known, the formula equations are algebraically rearranged as follows: $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$ $\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}$ $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$

Application Cases and Real-World Scenarios

The Law of Cosines applies specifically to two operational cases: Case 1 (SAS) and Case 2 (SSS).

Case 1 occurs when two side lengths and an included angle are given. The three possible combinations of known measures under Case 1 are:

Known sides $b$ and $c$ with included angle $A$ ( $c$ , $A$ , and $b$ )
Known sides $a$ and $c$ with included angle $B$ ( $a$ , $B$ , and $c$ )
Known sides $a$ and $b$ with included angle $C$ ( $a$ , $C$ , and $b$ )

A practical scenario for Case 1 (SAS) involves Maria and Nena, who are standing $10\,m$ away from each other on the ground. Between them is a stray balloon that is $16\,m$ away from Nena. If the angle of elevation from the ground forms $57^\circ$ , the distance from Maria to the stray balloon can be calculated. Because two sides ( $10\,m$ and $16\,m$ ) and the included angle ( $57^\circ$ ) are given, this represents an SAS case and is solvable by the Law of Cosines.

Case 2 occurs when all three side measures of an oblique triangle are given without any initial angle measurements. The known conditions under Case 2 are the three sides $a$ , $b$ , and $c$ .

A practical scenario for Case 2 (SSS) involves John and Sarah standing on different sides of a park $39\,m$ away from each other. Both individuals run towards a water fountain. John is $10\,m$ away from the fountain, while Sarah is $45\,m$ away from the same fountain. Relative to the fountain, the angle between John and Sarah can be determined. Because all three side distances ( $39\,m$ , $10\,m$ , and $45\,m$ ) are provided, this scenario represents an SSS case and can be solved using the Law of Cosines.

Assessment, Practice Problems, and Reference Standards

Formative Assessment #5 provides practical problems to evaluate understanding of triangle cases and the applicability of the Law of Cosines: Problem 1 through Problem 3 involve evaluating given triangle figures to state the case and confirm whether the Law of Cosines applies. Problem 4 states that in triangle $DEF$ , the measure of angle $D$ is $m\angle D = 14^\circ$ , side $e = 7\,m$ , and side $f = 4\,m$ . The task is to identify the case and solve for the missing side $d$ . Problem 5 states that in triangle $DEF$ , the measurements are $m\angle D = 30^\circ$ , side $f = 13\,ft$ , and side $d = 11\,ft$ . The task is to calculate the measure of side $e$

To solidify comprehension, real-life situations involving the Law of Cosines must be created for both Case 1 (SAS) and Case 2 (SSS). Each problem requires an accompanying structural illustration indicating the given measurements, labeling the target unknown values, and explicitly identifying the triangle case.

The curriculum content and mathematical guidelines are drawn from the following sources:

National Mathematics Program Lesson Scripts
Revised K-10 Budget of Work
chatgpt.com
Other learning resources
https://study.com/skill/practice/solving-a-word-problem-using-the-law-of-sines-questions.html