trigonometry
DESCRIPTION
trigonometry for class 10TRANSCRIPT
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*Trigonometry
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Instant TrigTrigonometry is math, so many people find it scaryIts usually taught in a one-semester high-school courseHowever, 95% of all the trig youll ever need to know can be covered in 15 minutesAnd thats what were going to do now
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Angles add to 180The angles of a triangle always add up to 180
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Right trianglesWe only care about right trianglesA right triangle is one in which one of the angles is 90Heres a right triangle:We call the longest side the hypotenuseWe pick one of the other angles--not the right angleWe name the other two sides relative to that angleHeres theright anglehypotenuseadjacentopposite
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The Pythagorean TheoremIf you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse adj2 + opp2 = hyp232 + 42 = 52, or 9 + 16 = 25 hyp = sqrt(adj2 + opp2)5 = sqrt(9 + 16)
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5-12-13There are few triangles with integer sides that satisfy the Pythagorean formula3-4-5 and its multiples (6-8-10, etc.) are the best known5-12-13 and its multiples form another set25 + 144 = 169
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RatiosSince a triangle has three sides, there are six ways to divide the lengths of the sidesEach of these six ratios has a name (and an abbreviation)Three ratios are most used:sine = sin = opp / hypcosine = cos = adj / hyptangent = tan = opp / adjThe other three ratios are redundant with these and can be ignoredThe ratios depend on the shape of the triangle (the angles) but not on the size
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Using the ratiosWith these functions, if you know an angle (in addition to the right angle) and the length of a side, you can compute all other angles and lengths of sidesIf you know the angle marked in red (call it A) and you know the length of the adjacent side, thentan A = opp / adj, so length of opposite side is given by opp = adj * tan Acos A = adj / hyp, so length of hypotenuse is given by hyp = adj / cos A
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Java methods in java.lang.Mathpublic static double sin(double a)If a is zero, the result is zeropublic static double cos(double a)public static double sin(double a)If a is zero, the result is zero However: The angle a must be measured in radiansFortunately, Java has these additional methods: public static double toRadians(double degrees)public static double toDegrees(double radians)
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The hard partIf you understood this lecture, youre in great shape for doing all kinds of things with basic graphicsHeres the part Ive always found the hardest:Memorizing the names of the ratios sin = opp / hypcos = adj / hyptan = opp / adj
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Mnemonics from wikiquoteThe formulas for right-triangle trigonometric functions are:Sine = Opposite / HypotenuseCosine = Adjacent / HypotenuseTangent = Opposite / AdjacentMnemonics for those formulas are:Some Old Horse Caught Another Horse Taking Oats AwaySaints On High Can Always Have Tea Or Alcohol
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You are at: (x, y)You want to move h units in the angle direction, to (x1, y1):So you make a right triangle...And you label it...And you compute: x1 = x + adj = x + hyp * (adj/hyp) = x + hyp * cos y1 = y - opp = y - hyp * (opp/hyp) = y - hyp * sin
This is the first point in your Turtle triangleFind the other points similarly...Drawing a Turtle
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The End
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