Trigonometry Instant Trig n n n Trigonometry is

Trigonometry

Instant Trig n n n 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 n And that’s what we’re going to do now

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

Right triangles n We only care about right triangles A right triangle is one in which one of the angles is 90° n Here’s a right triangle: Here’s the angle we are looking at hyp Here’s the ote nus right angle e opposite n n 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 n If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse n adj 2 + opp 2 = hyp 2 n 32 + 42 = 52, or 9 + 16 = 25 n n hyp = sqrt(adj 2 + opp 2) 5 = sqrt(9 + 16)

5 -12 -13 n n 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 hyp adj

n n 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: n n n 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 n opposite Ratios hyp ote nus e adjacent The ratios depend on the shape of the triangle (the angles) but not on the size hyp ote nus e 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 hyp ote nus e opposite n n adjacent If you know the angle marked in red (call it A) and you know the length of the adjacent side, then n n 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 n public static double sin(double a) n n n public static double cos(double a) public static double sin(double a) n n n 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 to. Radians(double degrees) public static double to. Degrees(double radians)

The hard part n n 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: n n n sin = opp / hyp cos = adj / hyp tan = opp / adj e s u en ot p y h adjacent opposite n Memorizing the names of the ratios

Mnemonics from wikiquote n The formulas for right-triangle trigonometric functions are: n n Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent Mnemonics for those formulas are: n n 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 (x 1, y 1): hyp You are at: (x, y) opp adj So you make a right triangle. . . And you label it. . . And you compute: x 1 = x + adj = x + hyp * (adj/hyp) = x + hyp * cos y 1 = y - opp = y - hyp * (opp/hyp) = y - hyp * sin This is the first point in your “Turtle” triangle Find the other points similarly. . .

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