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°
44°
68° 68°
20°
120°
30°
44°
68°
+ 68°
180°
20°
30°
180°
+ 130°
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:
• 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
right angle
hypotenuse
Here’s the angle
we are looking at
adjacent
opposite
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 its
multiples (6-8-10, etc.)
are the best known
• 5-12-13 and its multiples
form another set
• 25 + 144 = 169
hyp
adj
opp
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
• The ratios depend on the
shape of the triangle (the
angles) but not on the size
hypotenuse
adjacent
opposite 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
• 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
adjacent
opposite
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
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. You are at: (x, y)
You want to move h units in the
angle α direction, to (x1, y1):
So you make a right triangle...
And you label it...
hyp
opp
adj
And you compute:
x1 = x + adj = x + hyp * (adj/hyp) = x + hyp * cos α
y1 = y - opp = y - hyp * (opp/hyp) = y - hyp * sin α
This is the first point in your “Turtle” triangle
Find the other points similarly...
Drawing a “Turtle”
