Trigonometry and Applications REFERENCES HTTP EN WIKIPEDIA ORGWIKITRIGONOMETRY

Trigonometry and Applications REFERENCES: • HTTP: //EN. WIKIPEDIA. ORG/WIKI/TRIGONOMETRY

Outline 1. 2. 3. 4. 5. Vector motion (and an intro to vectors in general) Rotation / angles in 2 d. Polar => Rectangular coordinates Rectangular => Polar coordinates A little physics (enough for lab 8)

I. Motivation �Our character movements so far: �Other types of movement:

I. What is a (Rectangular 2 d) vector? �A collection of 2 values: pos = [400, 300] vel = [100, -30] # 100 px/s right, 30 px/s down �A position (pos) �An offset (vel) 100 -30

I. Law of similar triangles � 100 -30 b α c γ a β e β α T f γ U d

I. Law of Similar Triangles, cont. 100 � -30

I. The "new" type of movement �That's great, but we want asteroids-movement! �The "new" type of vector motion: Move n pixels (a distance) in this direction Q: How do we represent a direction? � A: In 2 d…an angle. � [In 3 d: quaternion, euler angle, direction vector, … <later>]

II. Angles (2 D) Two common systems: • Degrees • Radians By convention, 0 (degrees / radians) is to the right. A measure of rotation: • Negative is clockwise (by convention) • Positive is counter-clockwise (by convention) Also a description of orientation: • How much we've rotate from the 0 (right) position

II. Angles (2 D) Degrees 90 45 135 -180 0 360 180 270 -90 720

II. Angles (2 D) degrees, cont. �The number 360 is sort-of arbitrary Evenly divisible by a lot of numbers (2, 4, 8, …) Loosely based on #days/yr Babylonians used a sexagesimal number system (60 -based instead of our 10 -based system) �In the radians system, the number has a physical meaning…

II. Angles (2 D) radians �What is π? Common answer: 3. 14159… But what does it represent? ? ? �Definition of π… Circumference = 6. 283" Diameter = 2" Circumference = 1. 57" Diameter = 0. 5"

II. Angles (2 D) radians, cont. �

II. Angles (2 D) radians, cont. � d r θ

II. Angles (2 D) radians, cont. d π Let's say diam = 4 …the circumference would be 4π … halfway around would be 2π (d) …the radius is 2 …So the radian angle would be 2 π / 2 …π 0

II. Conversions �

II. Conversions, cont. �…Or just use the math functions. math. radians(num) � Converts num (assumed to be in degrees) to radians math. degrees(num) � Converts num (assumed to be in radians) to degrees �Caution: If you see the number 45. 0 in your code, is it in radians or degrees? � You can't tell – neither can python. Comments are very important!

II. Complementary Angles �A pair of complementary angles add up to 180 (degrees) If Θ and Φ are complementary… Θ + Φ = 180 �The complement of 34 degrees is 146 degrees.

II. Back to the original problem �Really – how do we move forward n pixels at an angle θ ? !? Cartesian Coordinates Angle (degrees) offset in x offset in y 0 n 0 90 0 n 180 -n 0 270 0 -n 45 ? ? 15. 4 ? ?

III. Trig to the rescue! � H O θ A

III. Trig functions � H O θ A A=H*cos(θ) H is the distance we want to move forward A is the amount to add to our x-position O=H*sin(θ) O is the amount to add to our y-position (note pygame's y axis) (A, O) is the Cartesian equivalent of (H, θ) in polar coordinate.

III. Polar => Cartesian conversion Back to our original problem… Initial assumption: distance is n (e. g. 15) This is the hypotenuse's length Cartesian Coordinates Angle (degrees) offset in x offset in y 0 1 0 90 0 1 180 -1 0 270 0 -1 15. 4 14. 46 n=15 3. 984 ? ? 15. 4⁰ ? ? 14. 46 3. 984 The length of the adjacent side's length (which we don't know)… …but we can calculate The opposite side's length this time A = H * cos(angle) O = H * sin(angle) = n * cos(15. 4) = n * sin(15. 4) = 14. 46 = 3. 984

III. Vectors �

III. Quadrants and Sign of trig functions �Let θ be any angle in the range –infininty…+infinity. �θ will be in one of 4 quadrants. �The following trig functions are positive in each quadrant: Q 1: Sin(θ), Cos(θ), Tan(θ) Q 2: Sin(θ) Q 3: Tan(θ) Q 4: Cos(θ) Quadrant III �Menmonic: "All Students Take Calculus" Quadrant IV

III. “Negative Distances” � Let's say our angle β is 130 degrees (Quadrant II) � Problem: We can't draw a right triangle with an (obtuse) angle β � We can, however, compute a complementary angle, α And then a right-triangle using that angle. � Notice how the adjacent side (if hyptonuse is 1) is cos(50) ≈ 0. 64 � This is the correct horizontal offset, but it is to the left of the origin. So…it really should be -0. 64. � Your calculator, pygame. math, etc, already handle this. � cos(130) ≈ -0. 64 � Interpret this as a distance of 0. 64, but to the left of the origin. β α 0. 64

III. Example �[Moving object, firing projectiles] [Add a "rotate-able" object]

IV. Rectangular => Polar �Why? Given: face-pos (fx, fy) and candy position (cx, cy) Find: angle to point the hand (towards candy) Estimate for this scenario: ~25 degrees [cx, cy] [fx, fy]

IV. Rectangular to Polar �We need to find: n: hypotenuse θ: the angle of the hand (at fx, fy) �Steps: [on board] [fx, fy] �Inverse Trig Functions: if sin(ψ) = a, sin-1(a) = ψ Similar for cos and tan. [continue solving on board] problems with just inverse trig functions Solution: math. atan 2(opp, adj) [cx, cy]

IV. Minimal amount of physics for the lab �