some general observations: math is not like a movie, or a novel, where you take it in "as a whole, in its entirety, at once". it's more like a chain: forged one link at a time (or maybe a jigsaw puzzle).

so you have to focus on "one step at a time", it takes some times before you can assemble the tinker-toys into "big structures".

some thoughts on trigonometry: everything is pythagoras: A^{2}+ B^{2}= C^{2}. this should remind you of the equation for a circle:

x^{2}+ y^{2}= r^{2}, because...well, it's the same underlying concept: the distance between a point on the circle, and the origin, IS the radius of the circle, which IS the length of the hypotenuse of the right triangle with vertical leg "y" and horizontal leg "x".

cosine is just a fancy name for all the possible x's we get, and sine a fancy name for all the y's we get on a circle. cosine and sine of..."what"? well, if we have a point on the circle, we have two lines:

the x-axis, and the line going through (0,0) and (x,y). so we can speak of the angle between these two lines. it turns out it's more convenient to consider the "half-lines" (rays) starting at (0,0), and considering the angle between the two rays (so we know which "V" of the "X" the two lines make we're talking about).

of course, if you're struggling with algebra, trig is going to be hell...because there is lots of "algebraic rearrangement" always going on.

here's the deal with algebra: it's just like arithmetic but "we don't know which numbers we have". what we DO have is the RULES of arithmetic to guide us. it's like this: if you owe me $10, how much do you have to pay me? $10. ok, let's just say you don't remember how much you owe me. what do you have to pay? "whatever you owe me".

that is:

X - X = 0 (where X = "whatever you owe me") <---one of the basic rules of arithmetic. also written as: X + (-X) = 0. this says something special about -X, and 0 (if you pay what you owe, you owe nothing).

so even though we don't KNOW (or remember) what "X" is, that doesn't mean we're totally "lost". we still know that what you owe, and what you pay should MATCH. so just the "rules themselves" give us SOME information, which we can sometimes use like a detective, to narrow down the possibilities.

of course, that's a "baby example"...but the rules are very powerful. for example, in grade school, one often learns "times tables":

1 times 6 is 6

2 times 6 is 12

3 times 6 is 18

and so on.

hmm...if one has already learned the "3's", one already knows that 6*3 is 18: 3+3+3+3+3+3+3 = 3+(3+3+3+3+3) = 3+(3+(3+3+3+3))

= 3+(3+(3+(3+3+3))) = 3+(3+(3+(3+(3+3)))) = 3+(3+(3+(3+6))) = 3+(3+(3+9)) = 3+(3+12) = 3+15 = 18

(because in learning the "times 3" we "count up by 3 each time" you can use your fingers...it's OK).

now if we know the RULE a*b = b*a, then we don't need to store 3*6 in our heads, just 6*3. this saves wear and tear on the old brain. so we can do this:

memorize a*1 for a = 0 through 9

memorize a*2 for a = 2 though 9 (since we already know a*1 = 1*a, and we've already covered it "in the 1's")

.....

memorize a*9 for a = 9 (although the "lower numbers of a" have already been covered).

all of the arithmetic we do "long-hand" is based on the SAME RULES we use to do algebra. here is another example:

(x+y)(x+y) = x*x + x*y + y*x + y*y = x^{2}+ 2xy + y^{2}(because...xy = x*y = y*x = so x*y + y*x = xy + xy = (1+1)(xy) = 2xy).

now that seems "abstract", but look:

(11)(11) = (10+1)(10+1) = 10*10 + 10 + 10 + 1 = 100 + 20 + 1 = 121. it's the same rules, it's just with "actual numbers" we're so used to "collecting the 100's, 10's and 1's" we forget that it's the RULES that let us actually DO that.

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unless you have some time-limit imposed upon you, feel free to take it as slow as you need until you UNDERSTAND. mathematics has a certain "organic" quality to it, the logic of it is designed for things 'to make sense". so if something is causing you trouble, don't "skip ahead" hoping it will "make sense later". untangle that knot, and lay the threads straight.