Make: Trigonometry: Build Your Way From Triangles to Analytic Geometry [1 ed.] 9781680457988

Table of contents :
Chapter 1. Trigonometry and Analytic Geometry
Definitions of trigonometry and analytic geometry
What algebra and geometry you should know already
Chapter 2. OpenSCAD and the 3D Printed Models
Downloading OpenSCAD
Downloading this book’s models
OpenSCAD capabilities
How to edit a pre-existing file in OpenSCAD
Chapter 3. Triangles and Trigonometry
Converting degrees to radians and vice versa
Proving that the angle of a triangle add to 180°
Using a protractor
Conventions for labeling triangle sides and angles
Acute, right, and obtuse triangles
Scalene, isosceles, and equilateral triangles
Understanding congruence and similarity of triangles
Applying similarity to the problem of measuring a large object
The Pythogorean Theorem
Square root notation
The Spiral of Theodorus as an analog square root calculator
Definition of basic trig functions: sine, cosine, tangent, secant, cosecant, cotangent, and their inverses
Chapter 4. Coordinate Systems and Analytic Geometry
Cartesian and polar coordinates
Coordinate of a point
Slope and intercept of a line
3D coordinates: Cartesian, cylindrical, spherical
Plotting a curve or surface in 3D
Chapter 5. The Unit Circle
Trig functions for angles above 90°
Principal value of an angle
The unit circle and continuous trig functions
Phase, frequency and amplitude of a trig function
Angular frequency
Period
Graphing trig functions in Cartesian and polar coordinates
Chapter 6. Trig Identities to Logarithms
Law of Sines and Law of Cosines
45-45-90 and 30-60-90 triangles
Trig cofunction relationships (cosecant, cosine, cotangent)
Complementary angles
Trig identity of squares of sine and cosine
Sum of angles and double-angle formulas
Prosthaphaeresis and transforming multiplication calculations into addition
Logarithms base 10
How a slide rule works
Calculating cube and square roots with logs
Operations with logs (multiplication, division, raising to a positive or negative power)
Estimation using log plots
Chapter 7. Navigation
Making and using an inclinometer
Measuring angles and distances with trig
Finding latitude by sighting Polaris
Trig applications in navigation using astrolabes, sextants, and GPS
Chapter 8. Making Waves
Nonsinusoidal periodic waves (sawtooth, square)
Superposition of waves
Constructive and destructive interference
Effects of wavelength and frequency on wave propagation
Point source model
Electromagnetic waves
Refraction and Snell’s Law
Rotated coordinate systems
Lenses and refraction
Reflection and Stokes’ Relations
Water and sound waves
Helicoid properties, as a minimal surface
Soap bubble demonstration of a minimal surface
Chapter 9. Ellipses and Circles
Conic sections
Circle, ellipse, parabola, and hyperbola cross-sections
Ranges of cut angle for generating each section
Ellipse properties: major/minor axes, foci, focal-distance sum relationship
Derivation of equations of ellipse and circle
Rotated and translated ellipses
Circumferences and areas of circle and ellipse
Application of ellipse: whispering gallery
Chapter 10. Parabolas and the Quadratic Formula
Slicing a cone to get parabolas
Focus and directrix of a parabola
FOIL method of multiplying algebraic expressions
Factoring polynomials
Parabola quadratic equations
Deriving the quadratic formula from completing the square
Vertex of a parabola
Computing the minimum or maximum
Parabola graph variation as parameters are varies
Translating parabolas
Parabolas in physics applications (parabolic reflectors, Galileo ramp)
The Fundamental Theorem of Algebra and computing roots
Chapter 11. Hyperbolas
Slicing two cones to get a hyperbola
Foci and circular directrix of a hyperbola
Derivation of standard equation from foci and directrix
Graphing a hyperbola with asymptotes
Reciprocal form of hyperbola
Hyperbolic trigonometry (sinh, cosh, tanh)
Application of hyperbolas (Cassegrain telescope)
Euler’s number and natural logs
Chapter 12. Applications and Looking Ahead
Tiling and tessellations
Tilings with regular polygons
Requirement for vertex angles in tilings to add to 360°
Aperiodic tilings, Penrose tilings, einstein monotiles
Using an Arduino processor and servos in a toy robot arm
Inverse kinematics of a simple robot arm
Converting Cartesian to spherical coordinates
Tangent function ambiguities and the atan2 function