Video summary

The Law of Sines

Main summary

Key takeaways

Educational

Main ideas / lessons

  • The Law of Sines extends triangle-solving beyond right triangles (where SOHCAHTOA works).
  • It applies to oblique triangles (triangles with no right angle).
  • It provides a consistent relationship between:
    • Angles (A, B, C) (capital letters)
    • Opposite sides (a, b, c) (lowercase letters), where each side is opposite its corresponding angle.

Core concept: Law of Sines

For an oblique triangle with angles (A, B, C) and opposite sides (a, b, c):

[ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} ]

Meaning: If you know some angles and/or sides, you can use this ratio to find the unknown sides/angles.

Methodology / workflow for solving triangles

General steps mentioned

  • If you know two angles, you can find the third using: [ A+B+C=180^\circ \quad \Rightarrow \quad \text{third angle} = 180^\circ - (\text{two known angles}) ]

  • Once you have an angle–side pair (an angle and the side opposite it), plug into the Law of Sines ratio to compute missing side lengths.

  • Angles are typically measured in degrees, and numeric answers are approximated using a calculator.

Case 1: Side–Angle–Angle (given one side and two angles)

Given: one side and two angles.

Steps:

  1. Compute the third angle: [ \text{third angle} = 180^\circ - (\text{angle}_1+\text{angle}_2) ]

  2. Use the known angle–opposite-side ratio: [ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} ]

  3. Solve for the other two side lengths using the corresponding sine relationships.

Case 2: Angle–Side–Angle (given two angles and the side between them)

Given: a side that lies between the two known angles (i.e., that side is opposite the third angle).

Steps:

  1. Find the third angle: subtract the two known angles from (180^\circ).

  2. Use the Law of Sines to plug in the angle-side information and solve for the remaining unknowns.

Case 3: Side–Side–Angle (given two sides and one angle)

Given: two side lengths and one angle.

Steps:

  1. Identify the angle opposite one of the known sides (the text frames this as “find the angle opposite this side length”).
  2. Subtract the known angles from (180^\circ) to get the third angle (when needed).
  3. Use Law of Sines to compute the third side length if required.

Special cases / viability checks:

  • Sometimes, the triangle cannot exist:
    • If solving produces (\sin(\text{angle}) > 1), that’s impossible, so the triangle can’t be formed.
  • Sometimes, there can be two solutions (an SSA ambiguity):
    • If the computed sine value corresponds to two different angles, then there may be two possible triangles.

Area formula for oblique triangles

  • Standard triangle area: [ \text{Area}=\frac{1}{2}(\text{base})(\text{height}) ]

  • If height is not directly known, use: [ \text{Area} = (\text{side}_1)(\text{side}_2)\sin(\text{included angle}) ]

  • Since an oblique triangle has three angles, the formula can be applied in three different ways (corresponding to choosing each angle as the included one between a pair of sides).

Speakers / sources featured

  • Professor Dave (speaker/host)

Original video