10-Apr-20

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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

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

 We only care about right triangles
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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 adjacent
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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

 If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse
 adj 2 + opp 2 = hyp 2
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3 2 + 4 2 = 5 2 , or 9 + 16 = 25
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 hyp = sqrt(adj 2 + opp 2 )
5 = sqrt(9 + 16)

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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 adjacent
 The ratios depend on the shape of the triangle (the angles) but not on the size adjacent

 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 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

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 public static double sin(double a)
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If a is zero, the result is zero public static double cos(double a) public static double sin(double a)
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If a is zero, the result is zero
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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)

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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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Memorizing the names of the ratios
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 sin = opp / hyp cos = adj / hyp tan = opp / adj adjacent

 The formulas for right-triangle trigonometric functions are:
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S ine = O pposite / H ypotenuse
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C osine = A djacent / H ypotenuse
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T angent = O pposite / A djacent
 Mnemonics for those formulas are:
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S ome O ld H orse C aught A nother H orse T aking O ats A way
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S aints O n H igh C an A lways H ave T ea O r A lcohol

You are at: (x, y)
You want to move h units in the angle
 direction, to (x
1
, y
1
): hyp 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
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This is the first point in your “Turtle” triangle
Find the other points similarly...