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Trigonometry and Vectors: Common Triangles and Vectors Review

Learn essential triangles, trigonometry, vectors with scalar and vector quantities, and components using engaging examples and diagrams.

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Trigonometry and Vectors: Common Triangles and Vectors 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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