Trigonometry is usually taught in high school and covers right triangle ratios. The key ratios are sine, cosine, and tangent, which relate the lengths of opposite, adjacent, and hypotenuse sides. The Pythagorean theorem states that the sum of the squares of the two shorter sides equals the square of the longest side. Trigonometric functions in Java like sin() and cos() require angles in radians. Memorizing the ratio definitions, like sine is opposite over hypotenuse, is considered the hardest part of trigonometry.

1. Trigonometry
Jul 9, 2018

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”

13. The End