Episode 27 Ladies and Gentlemen. Check it. I think I have said this before, but I really can't say it enough: I think the most daunting aspect of physics is that the foundations required to truly understand what's going on are never really solidly there, and so I am going to great lengths to make sure everything is going to be firmly in place before we move forward. One critical item that is glossed over time and time again in physics class is the importance of triangles and certain relations of triangles that are just accepted as known. Most of my friends in Physics I who were struggling weren't struggling because they were dumb. They would study and work very hard. They would go to office hours. Then when they revealed they were stuck, many times the piece of information they were missing was something that they never knew, and wasn't related in the textbook because it was assumed that the student walked in with that information! And many times, that information had to do with triangles and geometry. So here, for ep 27, are 27 must know facts about triangles.
0) Very simple but it needs to be stated: A triangle is formed by three lines and has three angles. It has three sides. Great!
1) The sum of the angles in any triangle are equal to 180 degrees (or pi radians).
2) Triangles come on four flavors, as far as we're concerned: Scalene, Equilateral (Equiangular), Isoceles, Right Triangles.
3) Scalene Triangles- These comprise most of the triangles out there. They have nothing particular properties about them (other than the 180 degree angle sum) that can be taken advantage of.
4) Isosceles Triangle- A triangle with minimum two equal sides. This gives several unique properties:
5) An Isoceles Triangle has minimum of two equal angles.
6) Any Isoceles Triangle can be broken into two right triangles by drawing a line bisecting the unequal angle. (More on this later)
7) Equilateral Triangle- An equiangular triangle or equilateral triangle has all sides of equal length and all angles at 60 degrees. This is calculated by dividing the total angle sum, 180, by three angles.
8) All equilateral triangles are isosceles, however not all isosceles triangles are equiangular. Why?
9) Right Triangle- Right triangles have one right angle, meaning an angle of 90 degrees. These types of triangles, for physicists, are likely the most interesting and have highly useful properties. This is generally because our coordinate systems are all orthogonal, aka all exist 90 degrees to each other.
10) The side which does not make an angle with the right angle is referred to as the hypotenuse.
11) The sum of the non-right angles in a right triangle are always equal to 90 degrees. If you know one non-right angle of a right triangle, calculating the other is simple.
12) Any side of a right triangle which is not the hypotenuse is referred to as a leg.
13) For a given angle, we can label the two legs with respect to this angle: the leg which forms the angle along with the hypotenuse is referred to as the adjacent side.
14) The side which DOES NOT form the angle is referred to as the opposite side.
15) Either leg can be the opposite or adjacent side depending on which angle you are referring to.
16) The lengths of the two legs of the triangle can be related to the hypotenuse by the pythagorean theorem: a^2 + b^2 = c^2 where a and b are the legs and c is the hypotenuse.
17) The angles of triangle can be calculated if specific leg lengths are known. Trigonometric functions sine, cosine, and tangent of the angles are used along with the leg lengths to calculate angle values.
18) To calculate the asymmetric angle of an isosceles triangle, it is easy to use right triangle rules with the bisected triangle to find "half the angle", and then double this value.
19) Special Right Triangle #1: 3-4-5 triangle: This triangle has sides which are proportional to the lengths 3-4-5. This is an example of a Pythagorean Triple, where all three sides of the triangle have whole number values. This makes for easy calculation if two of the sides are known.
20) Special Right Triangle #2: 45-45-90 Triangle: This triangle is an isosceles right triangle which has angles measuring 45, 45 and 90. In this type of triangle the sides share the proportion 1:1:sqrt(2). This is useful if the length of one of the legs of the triangle is known. In this triangle the lengths of both legs are equal.
21) Special Right Triangle #2: 45-45-90 Triangle Factoid #1: Either leg of the triangle is equal to one half the length of the hypotenuse multiplied by the square root of two. The one-half is due to keeping the radical value in the numerator.
22) Special Right Triangle #2: 45-45-90 Triangle Factoid #2: The hypotenuse of the right triangle is equal to the length of either leg multiplied by the square root of two.
23) Special Right Triangle #3: 30-60-90 Triangle: This triangle has angles measuring 30, 60 and 90. The sides of the triangle share the proportion 1:sqrt(3):2, where 2 is the length of the hypotenuse. This is useful if the length of one of the sides of the triangle is known.
24) Special Right Triangle #3: 30-60-90 Triangle Factoid #1: The length of the shorter leg is always equal to one half of the length of the hypotenuse.
25) Special Right Triangle #3: 30-60-90 Triangle Factoid #2: The length of the longer leg is always equal to one half of the length of the hypotenuse multiplied by the square root of three.
26) Special Right Triangle #3: 30-60-90 Triangle Factoid #3: The length of the longer leg is always equal to the shorter leg length multiplied by the square root of three.
27) Vector quantities can be described as related to their "component" vectors, which form a right triangle for coordinate axes which are orthogonal.