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Jul 27, 2011

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

44

68 68

20

12030

4468

+ 68

180

20

30

180

+ 130

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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:

We call the longest side the hypotenuse

We pick one of the other angles--notthe right angle

We name the other two sides relative to that angle

Heres the

right angle

hypotenuse

Heres the angle

we are looking at

adjacentoppos

it

e

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The Pythagorean Theorem

If you square the length of thetwo 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 its

multiples (6-8-10, etc.)

are the best known

5-12-13 and its multiples

form another set

25 + 144 = 169

hyp

adjopp

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Ratios

Since a triangle has threesides, 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

adjacentoppos

it

e

hypotenuse

adjacentoppos

it

e

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Using the ratios

With these functions, if you know an angle (in addition to theright 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

hypotenuse

adjacentopp

os

it

e

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Java methods in java.lang.Math

public static double sin(double a) Ifa is zero, the result is zero

public static double cos(double a)

public static double sin(double a) Ifa 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)

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The hard part

If you understood this lecture, youre in great shape fordoing all kinds of things with basic graphics

Heres the part Ive always found the hardest:

Memorizingthe names of the ratios

sin = opp / hyp

cos = adj / hyp

tan = opp / adj

hypote

nuse

adjacent

opp

os

it

e

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Mnemonics from wikiquote

The formulas for right-triangle trigonometric functionsare: Sine = Opposite / Hypotenuse

Cosine = Adjacent / Hypotenuse

Tangent = Opposite / Adjacent

Mnemonics for those formulas are: Some Old Horse Caught AnotherHorse Taking Oats Away

Saints On High Can Always Have Tea OrAlcohol

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You are at: (x, y)

You want to move h units in the

angle direction, to (x1, y1):

So you make a right triangle...

And you label it...

hypopp

adj

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...

Drawing a Turtle

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The End