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Agreed re: triangle method. At the very least, the triangle should be explored in the context of a unit circle. Otherwise, most students struggle with negativ
by candu 9y ago
Agreed re: triangle method. At the very least, the triangle should be explored in the context of a unit circle. Otherwise, most students struggle with negative sin / cos / tan values ("how do the quadrants work again?"), the connection to Cartesian coordinates, etc. It's a shame that we present so much of mathematics as arcane magic requiring incomprehensible mnemonics to understand - especially in trigonometry, where there are much more intuitive / visual representations of the basic ideas.
(Source: I volunteer with the Toronto Public Library to help high school students with homework, mainly math.)
- threatofrain 9y agoWhat's a top alternative you've found to work aside from triangles?
- meuk 9y agoFor me it was associating (sin(r), cos(r)) with a direction (on the unit circle), instead of with a triangle. After that, the relation with a triangle where the longest side has unit length is obvious.
- hexane360 9y agoDon't you mean (cos(r), sin(r))? I guess your method works if your angle starts at north and increases clockwise.
- rocqua 9y agoNot who you responded to, but as he stated, the best method is the unit circle. Take a circle of radius 1 around the origin. Draw a line with angle x through the origin. The point where this line intersects the circle has vertical position sin x and horizontal position cos x. Moreover, if you measure the angle in radians, x is how much of the unit circle your angle covers. This also immediately makes the identity (sin x)^2 + (cos x)^2 = 1 obvious. https://en.wikipedia.org/wiki/Unit_circle https://en.wikipedia.org/wiki/Unit_circle
- seanmcdirmid 9y agoWhen using sin and cos in graphical programming, I wish code completion for them came up in the context of a triangle and unit circle so I wouldn’t keep referring back to Wikipedia.
- Maybestring 9y agoIf, like me, you have the graph of the functions burned into your memory, then the anti-symmetry of sin makes it's geometric interpretation easy to remember. Sin and Cos correspond to the rejection and projection of two unit vectors. When they rotate past each other, the projection still has the same direction, but the rejection changes sign, so it must correspond to the anti-symmetric sin, not the symmetric cos.
- seanmcdirmid 9y agoI don't use them enough to completely internalize them, so having a reference when I do need to use them is very helpful.