Precalculus - The Law of Cosine and Area

Introduction

  • A triangle in which none of its interior angles are 90° is called a non-right triangle or oblique triangle.
  • Since trigonometric ratios can only be applied to right triangles, how can we solve non-right triangles? [Note that solving a triangle means finding the measurement of its angles and sides].
  • Here comes the role of the law of cosines. It relates the sides and angles of non-right triangles.

The Cosine Rule

  • Consider a triangle ABC as shown in the figure below:

  • The Law of Cosines or Cosine Rule states that:

a2=b2+c2-2bc·cos(A)

b2=a2+c2-2ac·cos(B)

c2=a2+b2-2ab·cos(C), where

a=the side opposite to angle A

b=the side opposite to angle B

c=the side opposite to angle C

  • The Cosine rule is applied under the following two conditions:

(a) when two sides and one included angle (SAS) is given, or

(b) when all three sides (SSS) are given.

Area of a Triangle

There are several ways to calculate the area of a triangle depending on the dimensions of the given triangle.

Case (1): Finding the area of a triangle when we know the measure of two of its sides and the included angle between them.

  • When the sides a and b and the included angle C are known, the area of the triangle can be calculated using the formula:

Area=12ab sin C

  • When the sides b and c and the included angle A are known, the area of the triangle can be calculated using the formula:

Area=12bc sin A

  • When the sides a and c and the included angle B are known, the area of the triangle can be calculated using the formula:

Area=12ac sin B

Case (2): Finding the area of a triangle when we know the measure of all its sides (Heron's Formula).

  • When the three sides of a triangle a, b, and c are known, the area of the triangle can be calculated using the formula:

Area=s s-as-bs-c, where s=a+b+c2 (semi-perimeter of the triangle)

Solved Examples

Example 1: In ABCAC= 10 cm, BC= 12 cm and cos C=34. Find the measure of side AB.

Solution: Here, two sides and included angle between them (SAS) are given. Let us draw the triangle based on the given data.

Applying the Cosine rule in the given triangle: 

c2=a2+b2-2ab·cos C

or, c2=122+102-2×12×10×34

or, c2=144+100-180=64 c=64=8

Therefore, the measure of side AB is 8 cm.

Example 2: In ABCAB= 8 cm, BC= 4 cm and AC= 6 cm. Find the measure of A.

Solution: Here, three sides are given (SSS). Let us first draw a triangle with the given data.

Applying the Cosine rule in the given triangle:

a2=b2+c2-2bc·cos A cos A=b2+c2-a22bc

or, cos A=62+82-422×6×8

or, cos A=36+64-162×6×8=8496=78

A=cos-17828.95°

Example 3: Find the area of the triangle shown in the figure.

Solution: Area=12ac sin B =12×15×12× sin 30° =45 cm2

Example 4: Find the area of the triangle shown in the figure.

Solution: a=22 cmb=16 cm, and c=18 cm

s=a+b+c2 =22+16+182 =28 cm

Area=ss-as-bs-c

=2828-2228-1628-18

=28×6×12×10

=2435 cm2

Cheat Sheet

  • For any triangle ABC (acute, obtuse, or right), if ab and c are the lengths of three sides opposite to A, B and C, respectively, then as per the law of Cosines or Cosine rule, the sides and angles of the triangle are related as mentioned below:

a2=b2+c2-2bc·cosA

b2=a2+c2-2ac·cosB

c2=a2+b2-2ab·cosC

  • When the law of cosines is applied to a right triangle, the result is the Pythagorean theorem.
  • The area of a triangle is half the product of the measure of any of its two sides times the sine of the included angle between the given sides.
  • When all the sides of a triangle are given, the area of the triangle can be computed using the following formula:

Area=ss-as-bs-c, where s=a+b+c2

Blunder Areas

  • The Laws of Cosines can be used in any triangle (not just right-angled triangles) except when there is ambiguity.