Law of Sines Summary

Pythagorean Theorem: $a^2 + b^2 = c^2$

Right Triangle:

• Contains a right angle (90°).

• Example: A right triangle has legs labeled as $a$ and $b$, and the hypotenuse as $c$.

Key Discoveries:

• Pythagoras (circa 500 BC) showed the relationship among right triangle sides.

• For any right triangle, if the legs are $a$ and $b$, the hypotenuse can be found with $c = ext{sqrt}(a^2 + b^2)$.

Practical Applications:

• Problem-solving with real-life scenarios:

• Distance calculations (e.g., driving coordinates, rectangles, baseball diamond).

• Example: Driving 48 miles west and then 36 miles south:

  • Use the theorem: $48^2 + 36^2 = c^2$

  • Result: $c = 60$ miles.

• Example: Diagonal of a rectangle with sides 15" and 8":

  • Apply theorem: $15^2 + 8^2 = c^2 <br>\Rightarrow c = 17"$.

Baseball Problem:

• Distance between bases is 90 feet; use theorem to find distance from home plate to second base.

Ladder Problem:

• Ladder leaning against a wall forms a right triangle. If ladder is 25 meters long and base is 7 meters from wall:

• Height of the window found via $c^2 = 25^2 - 7^2$.

Real-Life Problem Setup:

• Break down to right triangles and apply $a^2 + b^2 = c^2$ for finding unknown distances or sides.