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We already know so much about Vectors like…

We already know so much about Vectors like…. What is a Vector? What is a Scalar? What are the parts of a Vector? And wait…there’s more. Vector Components. Any vector can be broken up into two perpendicular components that lie along the x axis and along the y axis.

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We already know so much about Vectors like…

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  1. We already know so much about Vectors like… • What is a Vector? • What is a Scalar? • What are the parts of a Vector? • And wait…there’s more

  2. Vector Components • Any vector can be broken up into two perpendicular components that lie along the x axis and along the y axis. • We call these the x and y components of a vector. • You can think of an individual component of a vector as the shadow it would cast on either the x axis or the y axis. y component x component

  3. Components y component also y component x component

  4. Island of SOH-CAH-TOA [Exploring Trigonometric Functions]

  5. SOHCAHTOA Also Means: “Let’s do Trig!! But….What is Trigonometry! Trigonometry, or trig are sets of useful ratios used for finding various parts of right triangles.

  6. NOW…”Lets do Trig!”SOH-CAH-TOA [SOH] Sine – Opposite – Hypotenuse [CAH] Cosine – Adjacent – Hypotenuse [TOA] Tangent – Opposite - Adjacent

  7. First: A quick Geometry Review PARTS OF A RIGHT TRIANGLE OPPOSITE HYPOTENUSE ADJACENT

  8. So, Sohcahtoa![…trans. “Let’s do Trig.!”]

  9. We’ll break it into syllables.

  10. SOH-CAH-TOA

  11. SOH: SINE-OPP-HYP sin  = opposite side hypotenuse OPPOSITE HYPOTENUSE ADJACENT

  12. SOH-CAH-TOA

  13. CAH: COSINE-ADJ-HYP cos  = adjacent side hypotenuse OPPOSITE HYPOTENUSE ADJACENT

  14. SOH-CAH-TOA

  15. TOA: TAN-OPP-ADJ tan  = opposite side adjacent OPPOSITE HYPOTENUSE ADJACENT

  16. A QUICK REVIEW SOH Sin Opposite Hypotenuse CAH Cosine Adjacent Hypotenuse TOA Tangent Opposite Adjacent

  17. APPLICATIONS: Let’s solve a simple problem. Sohcahtoa!

  18. 25 m/s Vertical component Y- component 30 Horizontal component X- component The natives launch a coconut at 25 m/s at an angle of 30 with the horizontal. What are the horizontal and vertical components of the velocity?

  19. Vertical component Y- component opposite= sin* hypotenuse = sin 30 (25 m/s) = 0.5 (25 m/s) = 12.5 m/s 25 m/s 30 Horizontal component X- component adjacent = cos * hypotenuse = cos 30 (25 m/s) = 0.866 (25 m/s) = 21.7 m/s The solution!

  20. APPLICATIONS: Let’s try a WORD problem! Sohcahtoa!

  21. Find the Height of Mt. Sohcahtoa The volcano is rumbling. The Sohcahtoans need to sacrifice a Physics Student. They know the base of the mountain is 6 Km wide. The angle formed from the base to the top is 60°. How tall is the mountain? [Use your calculator!!!]

  22. Find the Height of Mt. Sohcahtoa What do we know? • = 60° Base = 6 Km = 3 Km 2 Height 6 Km

  23. What else do know? SOH - sin  = Opposite side Hypotenuse CAH - cos  = Adjacent side Hypotenuse TOA - tan  = Opposite side Adjacent side Which one should we use? Height 6 Km

  24. Which one? SOH - sin  = Opposite side Hypotenuse CAH - cos  = Adjacent side Hypotenuse TOA - tan  = Opposite side Adjacent side Height 6 Km

  25. WHY?? tan  = Opposite side Adjacent side tan 60° = h 3 Km Height h 3 Km

  26. WHY?? tan  = Opposite side Adjacent side tan 60°(3km) = h .(3km) 3 km (tan 60°)(3 Km) = h (0.866)(3) = h h = 2.598 Km Height h 3 Km

  27. WHY?? tan  = Opposite side Adjacent side tan 60°(3km) = h .(3km) 3 km (0.866 )(3 Km) = h answer h = 2.598 Km Height h= 2.598 3 Km

  28. EXCELLENT!!! You deserve a HAND! Now, Let’s Work!!! SOHCAHTOA!!!

  29. What if I want to find an angle? Here we know the opposite side and the hypotenuse. So we will use the sin function 20 cm 12 cm Sin-1(opp/hyp) =  - this is called the inverse sin function   = 36.87 degrees

  30. Back to the Island of SOHCAHTOA • The crafty physics student wants to escape death by volcano and land behind the angry SOHCAHTOANS. He builds a slide from the top of the Volcano (2.6 Km high) to land 7 Km from the base of the volcano. What are the angles the slide will make with the volcano and the ground?

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