Today's soundtrack is Primus: Tales From the Punchbowl, a really weird album with great musicianship.
Primus sucks btw.
This afternoon, I'm learning about the sine, cosine, and tangent functions of angles in standard position.
I started by watching this video as an introduction to the subject.
Let's start by looking at the sine-cosine functions.
When we're looking at an angle in standard position, we can draw a circle around it, note the point where the terminal arm intersects with the circle as point "P," draw a vertical line from P to y 0 to indicate one side of the triangle, and connect from that point to x 0 (thus, the origin of the graph) to complete our right triangle. We label the vertical line y, the horizontal line x, and the terminal arm r (because it is the radius of the Unit Circle that we drew). The distance from the origin to the circle (r) is one unit. This allows us to solve the triangle by using the Pythagorean theorem.
Now, given that we are working with a knowledge of angle theta (θ) touching the origin, we can say that cos θ = x/r, sin θ = y/r, and tan θ = y/x.
We can find the coordinates of P in a unit circle as long as we know the coordinates of A. All we need to do is is pythagoreally determine the ratios of the sides of the triangle. Let's say for example that we want to find the x-coordinate of an angle whose terminal arm goes to point A (7, -3). Which trigonometric function can we use to find x? Well, since x is our adjacent side, we will use the cosine function. COS = adj/hyp, which in this case is 7/r. But before we can find out the x-coordinate of P, we will use the Pythagorean theorem to determine the length of r when it lies one unit away from the origin. We know that our sides are 7 and -3, so we will solve for r in the equation 7² + -3² = r². 7² is 49; -3² is 9, so 49+9 = r². 49+9 = 58; 58 = r². We'll take the square root of both sides to isolate r, giving us r = √(58). We now know that COS θ = 7/√(58), which my calculator says is 0.92. So P has an x-coordinate of 0.92!
