Trigonometry 20-Apr-19

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

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

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: Here’s the angle we are looking at Here’s the right angle hypotenuse opposite 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 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)

hyp adj opp 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

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

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 hypotenuse adjacent opposite 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

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) 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 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

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

Drawing a “Turtle” You want to move h units in the angle  direction, to (x1, y1): hyp opp adj You are at: (x, y) 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