Trigonometry please (specifically Sin, Cos, Tan)! I can calculate the values when I need them, but I want to understand the 'why' and the 'how' on a fundamental level.
With trigonometry, Some key ideas that I like are:
sin, cos and tan refer to right-angle triangles (RAT) - it seems amazing that so much math can be based on the simple idea that we start with a triangle where one of the angles = 90 degrees;
sin, cos and tan are the ratios of two of the sides of a RAT: for example, cos(60) = 0.5 says that the ratio of the two arms of the 60 degree angle of the RAT is always 1:2. Saying it again: if you have a RAT with one angle = 60 degrees (the other angles are 30 & 90 degrees), one arm of the 60 degree angle = 1 unit and the other arm will be twice as long, that is 2 units (this second arm is the hypotenuse, the longest side, of the triangle).
The key idea is that sin, cos and tan are RATIOS. (The word similarity of RAT, an abbreviation I devised for this comment, and RATios has only just struck me.)
The ancient Egyptians were faced with a dilemma: the annual floods of the Nile destroyed the boundary markers of the farmers, so they had to come up with a means of surveying the land. The solution: the right-angle triangle.
Oh hey, I just got back to HN and saw this. Thanks for getting the simple concept of "RATios of sides" into my head! Inching closer to understanding :-)
Comments
Trigonometry please (specifically Sin, Cos, Tan)! I can calculate the values when I need them, but I want to understand the 'why' and the 'how' on a fundamental level.
http://betterexplained.com/articles/intuitive-trigonometry/ and the whole Better Explained website is pretty great for things like this. Their article on the constant e is what finally helped me to understand what e was: http://betterexplained.com/articles/an-intuitive-guide-to-ex...
cool! I will check it out. thank you
With trigonometry, Some key ideas that I like are:
sin, cos and tan refer to right-angle triangles (RAT) - it seems amazing that so much math can be based on the simple idea that we start with a triangle where one of the angles = 90 degrees;
sin, cos and tan are the ratios of two of the sides of a RAT: for example, cos(60) = 0.5 says that the ratio of the two arms of the 60 degree angle of the RAT is always 1:2. Saying it again: if you have a RAT with one angle = 60 degrees (the other angles are 30 & 90 degrees), one arm of the 60 degree angle = 1 unit and the other arm will be twice as long, that is 2 units (this second arm is the hypotenuse, the longest side, of the triangle).
The key idea is that sin, cos and tan are RATIOS. (The word similarity of RAT, an abbreviation I devised for this comment, and RATios has only just struck me.)
The ancient Egyptians were faced with a dilemma: the annual floods of the Nile destroyed the boundary markers of the farmers, so they had to come up with a means of surveying the land. The solution: the right-angle triangle.
Oh hey, I just got back to HN and saw this. Thanks for getting the simple concept of "RATios of sides" into my head! Inching closer to understanding :-)
For visual trig, there's a great interactive unit circle here:
http://www.touchmathematics.org/topics/trigonometry
Study the way the color-coded lengths vary as theta increases around the unit circle and you'll develop an intuitive sense of sin, cos, tan, etc.
Thanks for the link. I played around with it for a couple minutes and will be coming back to it later for sure.
This is from a fun thread that's been posted here before:
http://math.stackexchange.com/a/734790
That's a great idea.