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REVIEW

Trigonometry and Vectors. REVIEW. Common triangles in Geometry and Trigonometry. You must memorize these triangles. 45 o. 60 o. 2. 1. 1. 30 o. 45 o. 1. 2. 3. Trigonometry and Vectors. REVIEW. Common triangles in Geometry and Trigonometry. 5. 3. 4. 1. Trigonometry and Vectors.

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REVIEW

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  1. Trigonometry and Vectors REVIEW Common triangles in Geometry and Trigonometry You must memorize these triangles 45o 60o 2 1 1 30o 45o 1 2 3

  2. Trigonometry and Vectors REVIEW Common triangles in Geometry and Trigonometry 5 3 4 1

  3. Trigonometry and Vectors opposite hypotenuse sin A = opposite adjacent adjacent hypotenuse tan A = cos A = IOTPOLYENGINEERING 3-8 REVIEW Trigonometric Functions Trigonometric functions are ratios of the lengths of the segments that make up angles.

  4. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 Vectors • Scalar Quantities – a quantity that involves magnitude only; direction is not important • Tiger Woods – 6’1” • Shaquille O’Neill – 7’0” • Vector Quantities – a quantity that involves both magnitudeand direction How hard to impact the cue ball is only part of the game – you need to know direction too Weight is a vector quantity

  5. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 Scalar or Vector? • 400 mph due north • $100 • 10 lbs weight • 5 miles northeast • 6 yards • 1000 lbs force Magnitude and Direction Magnitude and Direction Vector Vector Magnitude only Magnitude only Scalar Scalar Magnitude and Direction Magnitude only Vector Scalar

  6. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 Vectors • Free-body Diagram • A diagram that shows all external forces acting on an object. applied force normal force N F Ff friction force force of gravity (weight) Wt

  7. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 Vectors • Describing vectors – • We MUST represent both magnitudeand direction. • Describe the force applied to the wagon by the skeleton: Hat signifies vector quantity 40 lbs 45o F = 40 lbs 45o magnitude direction

  8. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Scalar Multiplication • We can multiply any vector by a real number. • Original direction is maintained, new magnitude. 2 ½

  9. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Addition • We can add two or more vectors together. • Redraw vectors head-to-tail, then draw the resultant vector. • (head-to-tail order does not matter)

  10. March 14, 2011 IOTPOLYENGINEERING 3-10 Drill Find a + b Find 2 a y y a a b x x

  11. Drill IOTPOLYENGINEERING 3-10 Find 2 a y 2a a x

  12. Drill IOTPOLYENGINEERING 3-10 Find a + b y a+b a a b x

  13. Drill IOTPOLYENGINEERING 3-10 Find a + b b y a a+b b x

  14. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components • It is often useful to break a vector into horizontal and vertical components (rectangular components). • Consider the Force vector below. • Plot this vector on x-y axis. • Project the vector onto x and y axis. y F Fy x Fx

  15. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components This means: vector F = vector Fx + vector Fy Remember the addition of vectors: y F Fy x Fx

  16. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Unit vector Vectors – Rectangular Components Vector Fx= Magnitude Fxtimes vector i F = Fx i + Fyj Fx= Fx i i denotes vector in x direction y Vector Fy= Magnitude Fytimes vector j F Fy= Fy j Fy j denotes vector in y direction x Fx

  17. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components From now on, vectors on this screen will appear as bold type without hats. For example, Fx = (4 lbs)i Fy = (3 lbs)j F = (4 lbs)i + (3 lbs)j

  18. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components Each grid space represents 1 lb force. What is Fx? Fx = (4 lbs)i What is Fy? Fy = (3 lbs)j What is F? F = (4 lbs)i + (3 lbs)j y F Fy x Fx

  19. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components What is the relationship between q, sin q, and cos q? cos q = Fx / F Fx = F cos qi sin q = Fy / F Fy = F sin qj F Fy q Fx

  20. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components When are Fx and Fy Positive/Negative? Fy + Fy + y F Fx + Fx - F x F F Fx - Fx + Fy - Fy -

  21. IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components Complete the following chart in your notebook: I II III IV

  22. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components Each grid space represents 1 lb force. What is Fx? Fx = (-1 lbs)i What is Fy? Fy = (3 lbs)j What is F? F = (-1 lbs)i + (3 lbs)j y F x

  23. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) Break up all forces into x and y component forces 2) add up all of the component forces in x-direction 3) add up all of the component forces in y-direction 4) Write resultant as single vector in rectangular components 150 lb 60o 100 lb

  24. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 1) Break up all forces into x and y component forces Space Junk: 150 lb 60o

  25. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Break up all forces into x and y component forces Gravity 100 lb

  26. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 2) Add up all forces in x direction 150 lb 60o 100 lb

  27. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 3) Add up all forces in y direction 150 lb 60o 100 lb

  28. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 4) Write resultant as single vector in rectangular components 150 lb 60o 100 lb

  29. Classwork IOTPOLYENGINEERING 3-10 Complete Worksheet

  30. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction 2) sum of forces in y-direction 3) Write as single vector in rectangular components Fx = F cos Qi = (150 lbs) (cos 60) i = (75 lbs)i SFx= (75 lbs)i No x-component

  31. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction 2) sum of forces in y-direction 3) Write as single vector in rectangular components Fy = F sin Qj = (150 lbs) (sin 60) j = (75 lbs)j Wy = -(100 lbs)j SFy= (75 lbs)j - (100 lbs)j SFy = (75 - 100 lbs)j

  32. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction 2) sum of forces in y-direction 3) Write as single vector in rectangular components R = SFx +SFy R = (75 lbs)i + (75 - 100 lbs)j R = (75 lbs)i + (29.9 lbs)j

  33. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

  34. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

  35. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

  36. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

  37. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

  38. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

  39. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors CLASSWORK/ HOMEWORK Complete problem #4 on the Vector Worksheet

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