Trigonometry

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Trigonometry
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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

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Angles add to 180

• The angles of a triangle always add up to 180

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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

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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)

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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

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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

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

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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

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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)

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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

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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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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...
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The End