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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 “trig” you’ll ever need to know can be covered in 15 minutes And that’s what we’re going to do now. 20°. 44°. 30°. 120°. 68°.

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instant trig
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
angles add to 180

20°

44°

30°

120°

68°

68°

20°

44°

30°

68°

+ 130°

+ 68°

180°

180°

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

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

Here’s the angle

we are looking at

Right triangles

Here’s the

right angle

hypotenuse

opposite

adjacent

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)

The Pythagorean Theorem
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

hyp

opp

adj

5-12-13
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

hypotenuse

hypotenuse

opposite

opposite

adjacent

adjacent

Ratios
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 byopp = adj * tan A

cos A = adj / hyp, so length of hypotenuse is given byhyp = adj / cos A

hypotenuse

opposite

adjacent

Using the ratios
java methods in java lang math
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 amust be measured in radians
  • Fortunately, Java has these additional methods:
  • public static double toRadians(double degrees)
  • public static double toDegrees(double radians)
the hard part

hypotenuse

opposite

adjacent

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
  • cos = adj / hyp
  • tan = opp / adj
mnemonics from wikiquote
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
drawing a turtle

hyp

opp

adj

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...