I'm working on building tetris now in Java and am at the point of rotations…
I originally hardcoded all of the rotations, but found that linear algebra was the better way to go.
I'm trying to use a rotation matrix to rotate my pieces, and found I need a good understanding of trigonometry.
That being said, the way I visualize sin and cos is on the bottom part of the picture… a circle drawn on a graph… but images online show the sin cos relationship as the above picture… two waves that are "90 degrees" away from each other…
So, my questions are:
1) I can't seem to visualize how they are 90 degree apart. I can't see how those angles are formed in the above graph from sin and cos.
2) It's said that the relationship is Sin X = Cos (90 - x), and that sin is opposite/hypotenuse and cos = adjacent/hypotenuse… but what exactly does that mean? What do the quotients of these mean in relation to an angle? Let's say x = 45 degrees, adjacent = 5 inches, hypotenuse = 5 inches, and opposite = 4 inches… so sin would be 5"/9" = .55 inches… What does this .55 inches mean in relation to the angle? How is it helpful?
Thank you!!
Best Answer
In your $345$ triangle, you have $\sin x = \frac 35$ and $\cos x = \frac 45$. This gives $x \approx 36.87^\circ$. Can you see where the first two come from the diagram? The $90^\circ$ offset between sine and cosine comes from the fact that we measure horizontally for the cosine and vertically for the sine. While it is true that $\sin x = \cos (90^\circ -x)$ the $90^\circ$ offset is better expressed $\sin x = \cos x - 90^\circ$. This version has $x$ increasing in the same direction on both sides of the equation.
2) opposite is the side opposite the angle. In your diagram, the $3$ side is opposite angle $x$, the $4$ side is adjacent, and $5$ is the hypotenuse. As I said, this gives $\sin x=\frac 35 = 0.6$ It has no units. The $3$ and $5$ have units of length (say, inches) but when you divide them the quotient has no units. The trig functions result in unitless numbers. The $0.6$ gives you one way to construct the angle, though the fact that $\tan x = \frac 34$ is easier to see. You start going east $4$ units, then north $3$ units, and you have constructed the angle.