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Trigonometry

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Trigonometry 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 ... – PowerPoint PPT presentation

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Title: Trigonometry


1
Trigonometry
2
Instant Trig
  • Trigonometry is math, so many people find it
    scary
  • Its usually taught in a one-semester high-school
    course
  • However, 95 of all the trig youll ever need
    to know can be covered in 15 minutes
  • And thats what were going to do now

3
Angles add to 180
  • The angles of a triangle always add up to 180

4
Right triangles
  • We only care about right triangles
  • A right triangle is one in which one of the
    angles is 90
  • Heres a right triangle

Heres the right angle
hypotenuse
opposite
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
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
5-12-13
  • There are few triangles with integer sides that
    satisfy the Pythagorean formula
  • 3-4-5 and itsmultiples (6-8-10, etc.)are the
    best known
  • 5-12-13 and its multiples form another set
  • 25 144 169

7
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
  • The ratios depend on the shape of the triangle
    (the angles) but not on the size

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
  • 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
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
The hard part
  • If you understood this lecture, youre in great
    shape for doing all kinds of things with basic
    graphics
  • Heres the part Ive always found the hardest
  • Memorizing the names of the ratios
  • sin opp / hyp
  • cos adj / hyp
  • tan opp / adj

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
Drawing a Turtle
You want to move h units in theangle ?
direction, to (x1, y1)
You are at (x, y)
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 triangle
Find the other points similarly...
13
The End
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