1. Trigonometry
20-Apr-19

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. Angles add to 180° The angles of a triangle always add up to 180° 20°
120°
30°
44°
68°
20°
30°
180°
+ 130°
44°
68°
+ 68°
180°

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
we are looking at
Here’s 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
= 52, or = 25
hyp = sqrt(adj2 + opp2)
5 = sqrt(9 + 16)

6. hyp
adj
opp
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
and its multiples form another set
= 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
hypotenuse
adjacent
opposite
The ratios depend on the shape of the triangle (the angles) but not on the size
hypotenuse
adjacent
opposite

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
hypotenuse
adjacent
opposite
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)
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, 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
hypotenuse
adjacent
opposite

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 the angle  direction, to (x1, y1):
hyp
opp
adj
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