SOH CAH TOA. b c. sin opp. hyp. cos adj. hyp a c. tan opp. adj b a

Transcription

1 SOH CAH TOA sin opp hyp b c c 2 a 2 b 2 cos adj hyp a c tan opp adj b a

2 Trigonometry Review We will be focusing on triangles

3 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

4 Ex A: Given the following triangle a = 4.21u b = 7.43 u b c Angle C = 90.0 a What is the hypotenuse (c)?

5 Ex A: Given the following triangle c 2 = b 2 + a 2 c 2 = b c c = 8.54 u a How would you label the angles?

6 Same triangle A a = 4.21u b = 7.43 u c = 8.54 u b A c What is measure of smallest angle, θ A? a θ is the Greek letter theta and stands for angle

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

8 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

9 The ratios.. Sine = opposite / hypotenuse Cosine = adjacent / hypotenuse Tangent = opposite / adjacent These only work for right triangles! Show Table

10 Angle SinA CosA TanA

11 Naming the sides This side is opposite our angle. This is the longest side the hypotenuse. O H A right angled triangle This side is adjacent to our angle. A The angle we are interested in.

12 Naming the sides O H = Hypotenuse O = Opposite A = Adjacent

13 Naming the sides A H A O H O H A O H = Hypotenuse O = Opposite A H O A = Adjacent H A O

14 Ex B Consider this triangle. What is the Sine Ratio? O = 4cm 30 Opposite/Hypotenuse gives us the Sine Ratio. sin 30 = 4cm/8cm = 0.5. If you enter Sin 30 in your calculator you should get 0.5. Try it! (sin button is in the trig menu) If the opposite side was 6 cm, what would the hypotenuse be?

15 Ex C Consider this triangle. What is the angle? O = 5cm Name the sides in reference to the angle Determine which trig function to use To determine angle you use the inverse trig function for and enter the ratio of the corresponding sides. Sin -1 (5/12) = 24.62º θ Sin = O/H

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

17 Example A a = 4.21u b = 7.43 u c = 8.54 u A What is measure of smallest angle, θ A? b c Cos θ A = adj/hyp Cos -1 θ A (7.43/8.54) a θ A = 29.54

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

19 How would you determine the last angle B? The sum of all angles in a triangle equals 180º 180º - 90º º 60.46º

20 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

21 Ex D: A right triangle has a hypotenuse measuring 28.0 u. The smallest angle has a measure of What is the measure of side S? What is the measure of side T? What is the measure of the remaining angle? 22º T 28 u S

22 Ex D: A right triangle has a hypotenuse measuring 28.0 u. The smallest angle has a measure of What is the measure of side S? What is the measure of side T? What is the measure of the remaining angle? Label Sides What do you know? What function can you use to solve for opp? Opp = Sin Hyp Opp = (Sin22º)(28u) Hyp and angle Sin = Opp/Hyp 22º T adj Opp = 10.49u 28 u S opp

23 Ex D: A right triangle has a hypotenuse measuring 28.0 u. The smallest angle has a measure of What is the measure of side S? What is the measure of side T? What is the measure of the remaining angle? What function could you use to solve for side T? Adj = Cos Hyp A = Cos(22) 28 u 22º T adj a = 25.96u 28 u S u What is another way I could solve for this side? a 2 = c 2 - b 2 a 2 = (28u) 2 (10.49u) 2

24 Ex D: A right triangle has a hypotenuse measuring 28.0 u. The smallest angle has a measure of What is the measure of side S? What is the measure of side T? What is the measure of the remaining angle? How would you solve for the remaining angle *? Rename sides in reference to * What function would you use? How about Tan? Remember we are looking for an angle, not a ratio or side. Tan -1 (25.96/10.49) 22º Angle * = 68º T u opp 28 u * S u adj What s another way you can solve for angle*? Remember angles equal 180º 180º 90º - 22º

25 º θ 10m hyp 5m opp Θ??

26 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 = (sinθ)(hyp) (sin30º)(10m) = 5m?? opp 10m hyp 30º

27 Real World Applications

28 2.00 miles 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 40?

29 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 A H Putting our finger on H shows that H = A/C = 2.00 (cos 40 ) = = 2.61 miles

30 30.m 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. 500.m?

31 30 We can use the inverse tan to find the angle. = tan -1 (30m/500m) = 3.4 Ex F Finding An Angle (2) 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. T O A Putting our finger on T shows that tan = O/A

32 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

33 The Church Steeple? Step 2. Identify the sides. Here we have the Adjacent side and want to find the Opposite. So, we use the TOA triangle. T O A Putting our finger on O shows that O = T A = (tan (35º) 50m = m

34 Remember A C H O O S H T A SOH-CAH-TOA

35 Sin Finding the Opposite?? SOH-CAH-TOA O =? cm 30 S O H Opp = Sin Hyp = (Sin 30 ) 8 = 4 cm

36 Cos Finding the Adjacent?? SOH-CAH-TOA Adj 27 A =? km = Cos Hyp = (Cos 27 ) 12.3 = = 11.0 km A C H

37 Tan Finding the Opposite A = 16 cm 53?? SOH-CAH-TOA T O A Opp O =? cm = Tan Adj = (Tan 53 ) 16 = = 21 cm

38 Sin Finding the Hypotenuse?? SOH-CAH-TOA O = 87 m 36 S O H Hyp = Opp Sin = 87 (Sin 36 ) = = 150 m

39 Cos Finding the Hypotenuse 60 A = 0.80 cm?? SOH-CAH-TOA A C H Hyp = Adj Cos = 0.80 (Cos 60. ) = = 1.6 cm

40 Tan Finding the Adjacent?? SOH-CAH-TOA O = 3.1 cm Adj 30 A =? cm = Opp Tan = 3.1 (Tan 30. ) = = 5.4 cm T O A

41 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 Sin Cos Tan The opposite of these are SHIFT then Sin -1 Cos -1 Tan -1

42 Sin Finding the Angle??? SOH-CAH-TOA S O H O = 3.0 km Sin = Opp Hyp Sin = Sin = = Sin -1 (0.4285) = 25

43 Cos Finding the Angle??? SOH-CAH-TOA A = 12.1 cm A C H Cos = Adj Hyp Cos = Cos = = Cos -1 (0.834) = 33.4

44 Tan Finding the Angle O =??? SOH-CAH-TOA O 67.0 cm T A A = 187 cm Tan = Opp Adj Tan = Tan = = Tan -1 (0.358) = 19.7

45 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