Summary
Highlights
The video begins by stating the objectives: illustrating the Law of Cosines and solving for missing sides and angles. It recaps previously learned triangle combinations (AAS, ASA, SSA, SAS, SSS) and explains why the Law of Cosines is needed for SAS and SSS cases, as the Law of Sines is insufficient when there isn't an angle opposite a given side.
The Law of Cosines states that the square of any side of a plain triangle is equal to the sum of the squares of the other two sides diminished by twice the product of those two sides and the cosine of their included angle. Formulas are provided for finding missing sides (e.g., a² = b² + c² - 2bc cos A) and missing angles (e.g., cos A = (b² + c² - a²) / 2bc).
The steps for solving SAS triangles are outlined: first, find the side opposite the given angle using the Law of Cosines. Second, find the second angle (opposite the shorter of the two given sides) using the Law of Sines, which will always be acute. Third, find the third angle by subtracting the known angles from 180 degrees.
An example demonstrates solving an SAS triangle with b=1, c=3, and angle A=80 degrees. The missing side 'a' is calculated as 2.99 using the Law of Cosines. Then, angle B is found using the Law of Sines (19.2 degrees), and finally, angle C is determined by subtracting the found angles from 180 degrees (80.17 degrees).
Another SAS example is presented with an obtuse angle: n=7, m=12, and angle O=115 degrees. Side 'o' is calculated as 16.25 using the Law of Cosines. Angle N is then found using the Law of Sines (22.98 degrees), and angle M is calculated to be 42.02 degrees.
The steps for solving SSS triangles are detailed: first, determine if a triangle can be formed by checking if the sum of the two shorter sides is greater than the longest side. Second, find the angle opposite the longest side using the Law of Cosines to account for a potential obtuse angle. Third, solve for one of the remaining angles using the Law of Sines (it will be acute). Finally, find the third angle by subtracting the two known angles from 180 degrees.
An example of an impossible triangle is given with sides a=2, b=4, and c=8. Since 2+4=6, which is less than 8, no triangle can be formed.
An SSS triangle with sides e=11, f=20, and g=25 is solved. The sum of the two shorter sides (11+20=31) is greater than the longest side (25), confirming a triangle exists. Angle G is found to be 103 degrees 40 minutes (an obtuse angle) using the Law of Cosines. Subsequently, angle F is found using the Law of Sines (51 degrees 2 minutes), and angle E is calculated to be 25 degrees 18 minutes.
A practice problem is presented with sides a=5, b=8, and c=9. The sum of the two shorter sides (5+8=13) is greater than the longest side (9), confirming a triangle. Angle C is calculated as 84.26 degrees. Angle B is found using the Law of Sines (62.1 degrees), and angle A is determined to be 33.56 degrees. All angles are acute in this case.