Trigonometry
9-Jan-21
Instant Trig
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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
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And that’s what we’re going to do now
Angles add to 180°
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The angles of a triangle always add up to 180°
20°
44°
68°
44°
68°
+ 68°
180°
68°
30°
120°
20°
30°
+ 130°
180°
Right triangles
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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
opposite
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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
The Pythagorean Theorem
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If you square the length of the
two shorter sides and add
them, you get the square of the
length of the hypotenuse
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adj2 + opp2 = hyp2
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32 + 42 = 52, or 9 + 16 = 25
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hyp = sqrt(adj2 + opp2)
5 = sqrt(9 + 16)
5-12-13
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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
opp
adj
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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:
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sine = sin = opp / hyp
cosine = cos = adj / hyp
tangent = tan = opp / adj
The other three ratios are
redundant with these and can
be ignored
opposite
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opposite
Ratios
adjacent
The ratios depend on the
shape of the triangle (the
angles) but not on the size
adjacent
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
opposite
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adjacent
If you know the angle marked in red (call it A) and you know
the length of the adjacent side, then
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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
Java methods in java.lang.Math
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public static double sin(double a)
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public static double cos(double a)
public static double sin(double a)
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If a is zero, the result is zero
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)
The hard part
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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:
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sin = opp / hyp
cos = adj / hyp
tan = opp / adj
adjacent
opposite
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Memorizing the names of the ratios
Mnemonics from wikiquote
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The formulas for right-triangle trigonometric functions
are:
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Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Mnemonics for those formulas are:
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Some Old Horse Caught Another Horse Taking Oats Away
Saints On High Can Always Have Tea Or Alcohol
Drawing a “Turtle”
You want to move h units in the
angle  direction, to (x1, y1):
hyp
You are at: (x, y)
opp
adj
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...
The End