1. 9/29  Test Wednesday  Pick up review & calculator  Have Motion WS II out  Warm up 2: Look at the V v T graph below. Draw the D v T graph showing the same motion. V v T m/s sec Make sure you write date, question, & answer for warm ups. If you are absent you must get them from the back.

2. 10/1  Pick up calculator and trig notes.  Make sure calculator is in degree mode  Tue you handed in Motion WS II 1-12 & worked on Motion Review  Wed you took the Motion Test  Test corrections/retakes: Thur pm, Mon am & pm, Tue am ONLY  Warm Up #3  List 3 facts about triangles:

3. 10/5  Today you will finish Trig Packet  Tomorrow you will have a quiz  No warm up today  I will be here after school  11i) 0.633

4. 10/6 YESTERDAY YOU FINISHED TRIG PACKET. THERE WAS NOT A WARM UP. QUIZ TODAY Warm Up 4: Sketch the triangle We will solve together What do the 2 triangle have in common Solve that first 12.68 cm Now solve for X 12.98º 25º 30cm 55 cm X

5. 10/6 YESTERDAY YOU FINISHED TRIG PACKET. THERE WAS NOT A WARM UP. QUIZ TODAY Warm Up 4: Solve for X 25º 30cm 55 cm X

6. 10/7 YESTERDAY YOU HAD A QUIZ. QUIZ CAN BE MADE UP AFTER SCHOOL TODAY OR BEFORE SCHOOL ON THURSDAY  Pick up Homework Set (Empire State) This is due BOC Thursday  Today we will be going outside to do the survey lab. (After we finish ex F of the application notes)  NOW: Have your notes out.  Trig Test will be Friday Oct 16  Warm Up #5  What is the missing angle?  What can we say about sides A & B?  TURN IN WARM UPS TO SORTER 45º X B A

7. ANSWERS TO EMPIRE STATE PROBLEM SET  1. 36.45º 28.07º  2. 335.14 ft  3. 18831.6 ft  4. 278.11 ft

8.  This Monument is:

9. 10/8 HAVE HW OUT  Yesterday we did a survey Lab. See me to make it up. You also finished warm ups and turned them in.  Today: Pick up Vector Note Sheet.  I have duty this pm until 2:50  Trig Quizzes should have been made up yesterday or this morning.  HAPPY BIRTHDAY RANDI W!

10. What is the difference between these tools? We will be using a triangulation device like the bottom tool. Note that the center reference is 0º. This allows us to get the angle of elevation, not the zenith angle.

11. SURVEY LAB HOW WOULD I FIGURE OUT HEIGHT? This is same as reading on triangulation tool. If using a typical protractor, the angle would represent the zenith. This describes the angle of elevation 45º

12. SURVEY LAB HOW WOULD I FIGURE OUT HEIGHT? angle of elevation = 45º 45º This is same as elevation angle since 90-45-45 triangle

13. SOH CAH TOA

14. TRIGONOMETRY REVIEW  We will be focusing on triangles

15.  What is a right triangle?  A triangle with a 90º angle  What is a hypotenuse?  Side of right triangle opposite the 90º angle  What is Pythagoreans Theorem?  c 2 = a 2 + b 2 where c is the hypotenuse.  Only applies to right triangles

16. EX A: GIVEN THE FOLLOWING TRIANGLE a = 4.21u b = 7.43 u Angle C = 90.0° What is the hypotenuse (c) ? b a c

17. EX A: GIVEN THE FOLLOWING TRIANGLE c 2 = b 2 + a 2 c 2 = 4.21 2 + 7.43 2 c = 8.54 u How would you label the angles? b a c

18. SAME TRIANGLE A What is measure of smallest angle, θ A ? θ is the Greek letter theta and stands for angle b a c a = 4.21u b = 7.43 u c = 8.54 u A

19. SOH CAH TOA  This is a good time to review SOH CAH TOA  What does sine, cosine, and tangent?

20.  What does sine, cosine, and tangent represent?  The RATIO between given sides of a right triangle in reference to a specific angle.  SOH CAH TOA  Triangle Demo

21. THE RATIOS…..  Sine = opposite / hypotenuse  Cosine = adjacent / hypotenuse  Tangent = opposite / adjacent  These only work for right triangles!  Show Table

22. Angle SinA CosATanA

23. NAMING THE SIDES A right angled triangle The angle we are interested in. H This is the longest side — the hypotenuse. O This side is opposite our angle. A This side is adjacent to our angle.

24. NAMING THE SIDES H = Hypotenuse O = Opposite A = Adjacent O

25. NAMING THE SIDES H = Hypotenuse O = Opposite A = Adjacent H O A O H A H O A H O A H O A

26. EX B CONSIDER THIS TRIANGLE. WHAT IS THE SINE RATIO? 30° 4cm 8cm H = O = Opposite/Hypotenuse gives us the Sine Ratio. sin 30° = 4cm/8cm = 0.5. If the opposite side was 6 cm, what would the hypotenuse be? If you enter Sin 30 in your calculator you should get 0.5. Try it! (sin button is in the trig menu)

27. EX C CONSIDER THIS TRIANGLE. WHAT IS THE ANGLE? θ 5cm 12cm H = O = Name the sides in reference to the angle Determine which trig function to use Sin -1 (5/12) = 24.62º To determine angle you use the inverse trig function for and enter the ratio of the corresponding sides. Sin = O/H

28. Now go back to Example A and solve the angle using the inverse cosine function, then solve the angle using the inverse tan function

29. EXAMPLE A a = 4.21u b = 7.43 u c = 8.54 u What is measure of smallest angle, θ A ? Cos θ A = adj/hyp Cos -1 θ A (7.43/8.54) θ A = 29.54° a cb A

30. EXAMPLE A a = 4.21u b = 7.43 u c = 8.54 u What is measure of smallest angle, A? Tan θ = opp/adj Tan -1 θ A (4.21/7.43) θ A = 29.54° a b c A

31. HOW WOULD YOU DETERMINE THE LAST ANGLE B?  The sum of all angles in a triangle equals  180º  180º - 90º - 29.54º  60.46º

32. FOR RIGHT TRIANGLES  If you know any two sides, you can determine the angle  If you know a side and an angle other than 90, you can determine a side

33. EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U. THE SMALLEST ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE? 22º S T 28 u

34. EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U. THE SMALLEST ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE? 22º S T 28 u Label Sides opp adj What do you know? Hyp and angle What function can you use to solve for opp? Sin = Opp/Hyp Opp = Sin Hyp Opp = (Sin22º)(28u) Opp = 10.49u

35. EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U. THE SMALLEST ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE? 22º S T 28 u How would you solve for side T? 10.49 u adj c 2 = a 2 + b 2 I will call adjacent (T) side a and opposite (S) side b a 2 = c 2 - b 2 a 2 = (28u) 2 – (10.49u) 2 a = 25.96u

36. EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U. THE SMALLEST ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE? 22º S T 28 u How would you solve for the remaining angle * ? 10.49 u adj Remember angles equal 180º 180º – 90º - 22º Angle * = 68º *

37. Summary Putting it all together: If you need to determine an angle :  Name sides in reference to angle of interest  Determine formula  You know opp and hypotenuse, want θ :  sin -1 = (Opp/Hyp) Use inverse function  sin -1 (5m/10m)=30º 5m opp 10m hyp Θ ??

38. Summary Putting it all together: If you need to determine a side:  Name sides in reference to known angle  Determine formula You know angle and hypotenuse, want opposite: Opp = (sin θ)(Hyp) (sin30º)(10m) = 5m ?? opp 10m hyp 30º

39.  Real World Applications

40. EX E THE SWIMMER A swimmer attempts to swim due north to the pier 2.00 miles away but the current takes him at a bearing of 40°. After a while he notices he is due east of the pier. How far has he travelled? Step 1. Draw a diagram. pier 2.00 miles 40° ?

41. EX E THE SWIMMER ? 2 40° Step 2. Identify the sides. Here we have the Adjacent side and want to find the Hypotenuse. So we use the CAH triangle. C H A Putting our finger on H shows that H = A/C = 2.00 ÷ (cos 40°) = = 2.61 miles

42. EX F FINDING AN ANGLE (1) At Heathwick airport there is a forest just 500. m from the end of the runway. The trees can be as tall as 30. m. What is the minimum angle of climb if aircraft are to avoid the trees? Step 1. Draw a diagram. 30.m 500.m ?

43. EX F FINDING AN ANGLE (2) 30 500  Step 2. Identify the sides Here we have the Adjacent and Opposite sides and want to find an angle. So, we use the TOA triangle. Putting our finger on T shows that… tan  = O/A T A O We can use the inverse tan to find the angle.  = tan -1 (30m/500m)  = 3.4°

44. EX G THE CHURCH STEEPLE Eric decides to find the height of the steeple of his local church. He measures a distance of 50. m along the ground. The angle of elevation to the top of the steeple is 35°. How high is the steeple? Step 1. Draw a diagram. 50.m 35° ?

45. THE CHURCH STEEPLE ? 50 35° Step 2. Identify the sides. Here we have the Adjacent side and want to find the Opposite. So, we use the TOA triangle. Putting our finger on O shows that O = T × A = (tan (35º) × 50m = 35.01 m T A O

46. REMEMBER… S H O C H A T A O SOH-CAH-TOA

47. 30° ? cm 8 cm H = O = SH O SIN FINDING THE OPPOSITE SOH-CAH-TOA ?? Opp= Sin  × Hyp = (Sin 30°) × 8 = 4 cm

48. 27° ? km 12.3 km H = A = CH A COS FINDING THE ADJACENT SOH-CAH-TOA ?? Adj= Cos  × Hyp = (Cos 27°) × 12.3 = 0.891 × 12.3 = 11.0 km

49. 53° ? cm TA O TAN FINDING THE OPPOSITE O = A = 16 cm SOH-CAH-TOA ?? Opp= Tan  × Adj = (Tan 53°) × 16 = 1.327 × 16 = 21 cm

50. 36° 87 m ? m H = O = SH O SIN FINDING THE HYPOTENUSE SOH-CAH-TOA ?? Hyp= Opp  Sin  = 87  (Sin 36°) = 87  0.5878 = 150 m

51. 0.80 cm ? cm H = A = CH A COS FINDING THE HYPOTENUSE 60° SOH-CAH-TOA ?? Hyp= Adj  Cos  = 0.80  (Cos 60.°) = 0.80  0.50 = 1.6 cm

52. 30° 3.1 cm TA O TAN FINDING THE ADJACENT O = A = ? cm SOH-CAH-TOA ?? Adj= Opp  Tan  = 3.1  (Tan 30.°) = 3.1  0.5773 = 5.4 cm

53. WHAT HAPPENS WHEN YOU DON’T KNOW THE ANGLE? We can find the usable number mentioned previously using the ratios. The problem is we know need to convert it back into the original angle. The Buttons on your calculator are… SinCosTan The opposite of these are SHIFT then Sin -1 Cos -1 Tan -1

54. 3.0 km 7.0 km H = O = SH O SIN FINDING THE ANGLE  SOH-CAH-TOA ? ? ? Sin  = Opp  Hyp Sin  = 3.0  7.0 Sin  = 0.4285  = Sin -1 (0.4285)  = 25°

55.  12.1 cm 14.5cm H = A = CH A COS FINDING THE ANGLE SOH-CAH-TOA ? ? ? Cos  = Adj  Hyp Cos  = 12.1  14.5 Cos  = 0.834  = Cos -1 (0.834)  = 33.4 °

56.  67.0 cm TA O TAN FINDING THE ANGLE O = A = 187 cm SOH-CAH-TOA ? ? ? Tan  = Opp  Adj Tan  = 67.0  187 Tan  = 0.358  = Tan -1 (0.358)  = 19.7°

57.  If two vectors are not at right angles to each other then we must use the Law of Cosines:  C 2 = A 2 + B 2 – 2AB cos   “  ” or Theta, is any unknown angle but in this case it is the angle between the two vectors