I am autistic and have difficulty understanding certain things for (sometimes) no apparent reason. This is one of those times. I am in need of help, and would appreciate some explanation. Thankyou! Please simplify it as much as possible.
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There is a really useful thing with triangles. If the angles are the same, the lengths of the sides are in proportion. This can be used in the real world to work out things like the height of a tree or a mountain.
So it comes in handy to have a shortcut of all possible triangles, because you can then use what information you have to figure out the rest. For example, you have the distance between yourself and the base of the tree, and the angle looking from the ground to the top of the tree. You can use a reference triangle with the same angle to work out the height of the tree.
To make a set of all the possible reference triangles, a Unit Circle works well. This is a circle with radius one, drawn on a Cartesian plane, which has a horizontal axis and a vertical axis. To make a unit circle, imagine an arrow drawn on a blank Cartesian plane. It starts completely flat (lying on the horizontal axis). The arrow starts at zero and the tip is exactly at one. If you keep the origin of the arrow at zero, and slowly rotate the arrow anti clockwise, the angle between the arrow and the horizontal axis will increase and over a full rotation will draw a circle with radius one.
At each step of the rotation, you can make a triangle, showing the width (along the horizontal axis) and height (along the vertical axis). You can use Pythagoras’s theorem to work out these numbers. The height is called Sine and the width Cosine.
Imagining the arrow lying on the horizontal axis. This is an angle of zero. The height, or Sine, is zero. The Cosine, one. As the arrow spins anticlockwise, the height goes from zero to a maximum of one, before decreasing back to zero, then decreasing to negative one, then up to zero as the arrow swings right around a full rotation. At the same time for the width, or cosine, starts at one, goes to zero, then foes to negative one, then back to zero, then back to one. Since the arrow can keep spinning around forever (like a backwards clock), the angle can go on forever, which is why the graph of sine and cosine look like waves that go up and down forever.
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