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Trigonometry

Trigonometry. 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. 20°. 44°. 30°. 120°. 68°.

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Trigonometry

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  1. 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. 20° 44° 30° 120° 68° 68° 20° 44° 30° 68° + 130° + 68° 180° 180° Angles add to 180° • The angles of a triangle always add up to 180°

  4. 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 angle we are looking at Right triangles Here’s the right angle hypotenuse opposite adjacent

  5. 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) The Pythagorean Theorem

  6. There are few triangles with integer sides that satisfy the Pythagorean formula 3-4-5 and itsmultiples (6-8-10, etc.)are the best known 5-12-13 and its multiples form another set 25 + 144 = 169 hyp opp adj 5-12-13

  7. 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 hypotenuse hypotenuse opposite opposite adjacent adjacent Ratios

  8. 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 byopp = adj * tan A cos A = adj / hyp, so length of hypotenuse is given byhyp = adj / cos A hypotenuse opposite adjacent Using the ratios

  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 amust be measured in radians • Fortunately, Java has these additional methods: • public static double toRadians(double degrees) • public static double toDegrees(double radians)

  10. hypotenuse opposite adjacent 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

  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. hyp opp adj Drawing a “Turtle” You want to move h units in theangle  direction, to (x1, y1): 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...

  13. The End

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