24 trigonometry(1)


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24 trigonometry(1)

  1. 1. Trigonometry Jun 30, 2012
  2. 2. Instant Trig 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
  3. 3. Angles add to 180° The angles of a triangle always add up to 180° 20° 44° 30° 68° 68° 120° 20° 44° 30° 68° + 130° + 68° 180° 180°
  4. 4. Right triangles We only care about right triangles  A right triangle is one in which one of the angles is 90°  Here’s a right triangle: Here’s the angle hyp we are looking at Here’s the opposite ote right angle nus e adjacent We call the longest side the hypotenuse We pick one of the other angles--not the right angle We name the other two sides relative to that angle
  5. 5. The Pythagorean 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 + opp2 = hyp2 32 + 42 = 52, or 9 + 16 = 25 hyp = sqrt(adj2 + opp2) 5 = sqrt(9 + 16)
  6. 6. 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.) hyp opp are the best known adj 5-12-13 and its multiples form another set 25 + 144 = 169
  7. 7. Ratios Since a triangle has three hyp opposite sides, there are six ways to ote n use divide the lengths of the sides Each of these six ratios has a adjacent name (and an abbreviation) Three ratios are most used:  The ratios depend on the  sine = sin = opp / hyp shape of the triangle (the  cosine = cos = adj / hyp angles) but not on the size  tangent = tan = opp / adj opposite h yp The other three ratios are ote nus e redundant with these and can adjacent be ignored
  8. 8. Using 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 hyp opposite ote nus e adjacent  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
  9. 9. Java methods in java.lang.Math 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)
  10. 10. The hard part 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 use opposite en hy pot cos = adj / hyp tan = opp / adj adjacent
  11. 11. 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
  12. 12. Drawing a “Turtle” You want to move h units in the angle α direction, to (x1, y1): hyp oppYou are at: (x, y) adj So you make a right triangle... And you label it... And you compute: x1 = x + adj = x + hyp * (adj/hyp) = x + hyp * cos α y1 = y - opp = y - hyp * (opp/hyp) = y - hyp * sin α This is the first point in your “Turtle” triangle Find the other points similarly...
  13. 13. The End