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Welcome to Week 2 College Trigonometry

Welcome to Week 2 College Trigonometry. Radians. You have probably always measured angles in degrees Radians are another way of measuring angles used in CAD and by engineers (blame them…). Radians. It was probably invented by an Arabic mathematician Jamshid al- Kashi in about 1400.

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Welcome to Week 2 College Trigonometry

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  1. Welcome to Week 2 College Trigonometry

  2. Radians You have probably always measured angles in degrees Radians are another way of measuring angles used in CAD and by engineers (blame them…)

  3. Radians It was probably invented by an Arabic mathematician Jamshid al-Kashi in about 1400

  4. Radians We can blame it on Roger Cotes a British mathematician and Newton’s proofreader who in 1714 introduced the idea to the English-speaking world

  5. Radians It uses the fact that the circum- ference of a circle is 2π*r

  6. Radians Suppose you have a unit circle where r = 1

  7. Radians To convert from degrees to radians, multiply the degrees by: π radians 180°

  8. RADIANS IN-CLASS PROBLEMS Convert from degrees to radians: 0° 180° 45° 270° 90° 360° 330° 96° 77° 11°

  9. Radians To convert from radians to degrees, multiply the radians by: 180° . π radians

  10. RADIANS IN-CLASS PROBLEMS Convert from radians to degrees: 0 π/2 π2π 3π/2π/4 π/73π/13 10π/9 9π/10

  11. Radians Radians will often have a “π” to let you know you are dealing with radians

  12. Radians Radian measure on your calculator – make sure you change from “Deg” to “Rad”

  13. RADIANS IN-CLASS PROBLEMS Find the trig values: sin(π/4) cos(π/4) tan(π/4) sin(π/12) cos(3π/5) tan(5π/7)

  14. Radians The radian angle, θ, can be found using the following formula: θ = radians where s is the length of the circle’s arc and ris the circle’s radius

  15. Radians The length of the circular arc s is: r θ

  16. Radians Linear and angular speeds are used to describe motion on a circular path The linear speed “v” is calculated in terms of the arc length: v = s = rθ t = time in seconds

  17. Radians The angular speed “w” is calculated in terms of angle θ w = θ= angle measured in radians t = time in seconds

  18. Questions?

  19. Graphs of Trig Functions Periodic Functions – repeat the pattern over and over

  20. Graphs of Trig Functions How to recognize which function is which: The shape, intercepts and asymptotes are critical

  21. Graphs of Trig Functions The sine function was first described by an Indian mathematician Aryabhata in the 5th century

  22. Graphs of Trig Functions Coffee stirrer

  23. Graphs of Trig Functions

  24. Graphs of Trig Functions It undulates back and forth It has a value of 0 at 0°

  25. Graphs of Trig Functions

  26. Graphs of Trig Functions It undulates back and forth It doesn’t have a value of 0 at 0°

  27. Graphs of Trig Functions Does it wiggle back and forth? It is a sin or cos Does it look “weird”? It’s not

  28. Graphs of Trig Functions What is it doing at x = 0? If y = 0 at x = 0, it is a sin If y = something else at x = 0, it is a cos

  29. TRIG GRAPHS IN-CLASS PROBLEMS Identify sines or cosines:

  30. Graphs of Trig Functions Sines and cosines are useful because of the shape of their curves Many natural phenomena undulate back and forth

  31. Graphs of Trig Functions

  32. Graphs of Trig Functions

  33. Graphs of Trig Functions A lot of mathematical models for real-world phenomena have a “sine” or “cosine” in them to include this cyclical pattern

  34. Graphs of Trig Functions

  35. Graphs of Trig Functions A tangent isn’t important because it is a useful graph shape for mathematical modeling A tangent is useful because it is a slope ()

  36. Graphs of Trig Functions

  37. Graphs of Trig Functions Ditto cotangent It’s important because it is 1/slope (Not really…)

  38. Graphs of Trig Functions

  39. Graphs of Trig Functions The secant is used in calculus

  40. Graphs of Trig Functions

  41. TRIG GRAPHS IN-CLASS PROBLEMS Which function?

  42. Graphs of Trig Functions I’ve never seen the cosecant used outside of a trig class… So… why do we study 6 trig functions rather than just 3?

  43. Graphs of Trig Functions

  44. Questions?

  45. Functions Graphs of common algebraic functions: constant identity abs value quadratic square root cubic cube root

  46. Transforming Functions vertical shifts: f (x) + c plus shifts up f (x) - c minus shifts down

  47. Transforming Functions vertical stretch: f (x) *c stretches f (x) ÷ c squashes

  48. Transforming Functions Any operations done to the function outside of the parentheses affects the “y” values These operations lead to an intuitive result

  49. Transforming Functions horizontal shifts: f (x + c) shifts left f (x - c) shifts right

  50. Transforming Functions horizontal shifts: f (x * c) squashes f (x ÷ c) stretches

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