basic description to main three trigonometric functions
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Trigonometry 23-Sep-17
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Instant Tri Trigonometry is math, so many people find it scary It's usually taught in a one-semester high-school course However, 95% of all the "trig" you'll ever need to know can be covered in 15 minutes And that's what we're going to do now
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An les add to 1800 The angles of a triangle always add up to 1800 680 680 + 680 1800 680 200 300 1200 200 300 1800
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Ri ht trian les We only care about right triangles A right triangle is one in which one of the angles is 900 Here's a right triangle: Here's the right angle ad •acent We call the longest side the hypotenuse Here's the angle we are looking at We pick one of the other angles--not the right angle We name the other two sides relative to that angle
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The P tha orean Theorem • If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse adj2 + opp 2 = hyp2 32 + 42 = 52, or 9 + 16 = 25 hyp = sqrt(adj2 + opp2) 5 = sqrt(9 + 16)
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5-12-13 There are few triangles with integer sides that satisfy the Pythagorean formula 3-4-5 and its multiples (6-8-10, etc.) are the best known 5-12-13 and its multiples form another set 25 + 144 = 169 opp hyp ad' oooooooooooo oooooaooooao oooooaooooao
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Ratios Since a triangle has three sides, there are six ways to divide the lengths of the sides Each of these six ratios has a name (and an abbreviation) Three ratios are most used: • sine = sin = opp / hyp cosine = cos = adj / hyp • tangent = tan = opp / adj The other three ratios are redundant with these and can be ignored ad •acent The ratios depend on the shape of the triangle (the angles) but not on the size hypotenuse ad •acent
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Usin the ratios With 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 sides ad •acent If you know the angle marked in red (call it A) and you know the length of the adjacent side, then • tan A = opp / adj, so length of opposite side is given by opp = adj * tan A cos A = adj / hyp, so length of hypotenuse is given by hyp = adj / cos A
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Java methods in public static double sin(double a) • If a is zero, the result is zero public 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 radians Fortunately, Java has these additional methods: public static double toRadians(double degrees) public static double toDegrees(double radians)
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The hard art If you understood this lecture, you're in great shape for doing all kinds of things with basic graphics Here's the part I've always found the hardest: Memorizing the names of the ratios sin = opp / hyp adj / hyp • cos = opp / adj tan = ad •acent
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Mnemonics from wikiquote The formulas for right-triangle trigonometric functions are: • Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent Mnemonics for those formulas are: • Some Old Horse Caught Another Horse Taking Oats Away • Saints On High Can Always Have Tea Or Alcohol
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Drawin a "Turtle" You want to move h units in the angle u direction, to (Xl, YD: hyp OPP are at: (x, y) Yo ake a right triangle... So yo you label it... And you compute: xl = x + adj = x + hyp * (adj/hyp) = x + hyp * cos u yl = y - opp = y - hyp * (opp/hyp) = y - hyp * sin u This is the first point in your "Turtle" triangle Find the other points similarly...
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The End
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