21 trigonometry

1. 11-Dec-21
Trigonometry

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
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
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
 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
adjacent
opposite
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
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 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 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  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 
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”

13. The End