1. SOHCAHTOA passport to Trigonometry Land... CLICK TO CONTINUE

2. Oops! Not a triangle!  I am sorry. I am not a triangle! Click to Review & Try Again!

3. I am a Triangle! Good job! You know me! Click: I’ll take you BACK

4. What is a triangle? A triangle is a polygon made up of three connected line segments in such a way that each side is connected to the other two. CLICK TO CONTINUE

5. Examples of triangles... All these polygons are tri-gons and commonly called triangles CLICK TO CONTINUE

6. These are not triangles... None of these is a triangle... Can you tell why not? CLICK TO CONTINUE

7. Try it yourself... Click on each one that IS a triangle? CLICK TO CONTINUE

8. Right triangles A right triangle is a special triangle that has one of its angles a right angle. You can tell it is a right triangle when when one angle measures 90 0 or the right angle is marked by a little square on the angle whose measure is 90 0. CLICK TO CONTINUE

9. These are right triangles... CLICK TO CONTINUE All right triangles

10. These are not right triangles... These triangles are NOT right triangles. Explain why not? CLICK TO CONTINUE

11. Try it yourself... Click on the triangle that is NOT a right triangle? CLICK TO CONTINUE

12. I am a right triangle  I am sorry you did not recognize me as one. I am a right triangle! Click to Review & Try Again!

13. Correct… That is not a right Triangle Well done! You know your right triangles well! Click: I’ll take you BACK

14. Hypotenuse The longest side of a right triangle is the hypotenuse. The hypotenuse lies directly opposite the right angle. The legs may be equal in length or one may be longer than the other. CLICK TO CONTINUE

15. Parts of a right triangle... A right triangle has two legs and a hypotenuse... hypotenuse leg CLICK TO CONTINUE

16. Try it yourself … hypotenuse Click on the side that is the hypotenuse of the right triangle. side 2 side 1 side 3 CLICK TO CONTINUE

17. Try it yourself … shorter leg Click on the side that is the shorter leg of the right triangle. side 2 side 1 side 3 CLICK TO CONTINUE

18. Try it yourself … longer leg Click on the side that is the longer leg of the right triangle. side 2 side 1 side 3 CLICK TO CONTINUE

19. Correct – Way to go! Excellent! Wise choice. You know your parts! Click: Go BACK and CONTINUE

20. Incorrect Choice!  I am sorry. You chose the wrong side! CliCk here to Review & Try Again!

21. Pythagorean Theorem... The right triangle has a special property, called the Pythagorean Theorem, that can help us find one side if we know the other two sides. hypotenuse leg 1 leg 2 c a b If the lengths of hypotenuse and legs are c, a and b respectively, then c 2 = a 2 + b 2 CLICK TO CONTINUE

22. Finding a side - hypotenuse Use the Pythagorean Theorem to find the length of the missing side. 14 10 x c 2 = a 2 + b 2 x 2 = 10 2 + 14 2 = 100 + 196 = 296 x= sqrt(296) = 17.2 CLICK TO CONTINUE

23. Finding a side … leg Use the Pythagorean Theorem to find the length of the missing side. x 10 15 c 2 = a 2 + b 2 15 2 = 10 2 + x 2 225= 100 + x 2 x 2 = 125 x= sqrt(125) = 11.18 CLICK TO CONTINUE

24. Try it yourself- hypotenuse Find the hypotenuse of the given right triangle with the lengths of the legs known: x 8 6 Click on the selection that matches your answer: A. 36 C. 100 B. 10 D. 64 CLICK TO CONTINUE

25. Try it yourself… leg Find the leg of the given right triangle with the lengths of the leg and hypotenuse known: 15 9 x Click on the selection that matches your answer: A. 24 C. 144 B. 6 D. 12 CLICK TO CONTINUE

26. Correct Answer… Great job! You take after Pythagoras! Click: Go BACK and CONTINUE

27. Oops … not quite!  I am sorry. Check your calculations again. Click to Review & Try Again!

28. Opposite or Adjacent side…? In a right triangle, a given leg is called the adjacent side or the opposite side, depending on the reference acute angle. hypotenuse leg 1 leg 2 c a b A Adjacent or opposite from an acute reference angle refers only to legs and not the hypotenuse CLICK TO CONTINUE

29. Opposite side (to an acute angle) In a right triangle, a given leg is called the adjacent side or the opposite side, depending on the reference acute angle. hypotenuse leg 1 leg 2 c a b A leg 2 is opposite to acute angle A leg 1 is NOT opposite to acute angle A CLICK TO CONTINUE

30. Adjacent side (to an acute angle) In a right triangle, a given leg is called the adjacent side or the opposite side, depending on the reference acute angle. leg 1 is adjacent to acute angle A leg 2 is NOT adjacent to acute angle A hypotenuse leg 1 leg 2 c a b A CLICK TO CONTINUE

31. Try it yourself … opposite Click on the side that is opposite to angle B. hypotenuse leg 1 leg 2 c a b B CLICK TO CONTINUE

32. Try it yourself … adjacent Click on the side that is adjacent to angle B. hypotenuse leg 1 leg 2 c a b B CLICK TO CONTINUE

33. Correct reference! Great job! You understood these references! Click: Go BACK and CONTINUE

34. Not exactly…  I am sorry. Opposite is across from the angle. Adjacent is next to the angle. Hypotenuse is neither adjacent nor opposite. Click to Review & Try Again!

35. Trigonometric ratios of an acute angles of a right triangle The ratios of the sides of a right triangle have special names. There are three basic ones we will consider: sine cosine tangent CLICK TO CONTINUE

36. The sine of an acute angle... Let the lengths of legs be a and b, and the length of the hypotenuse be c. A is an acute angle. hypotenuse leg 1 leg 2 c a b A With reference to angle A, the ratio of the length of the side opposite angle A to length of the hypotenuse is defined as: sine A = = a/c Sine A is abbreviated Sin A. Thus, sin A = a/c. length of side opposite  A length of the hypotenuse CLICK TO CONTINUE

37. The cosine of an acute angle... Let the lengths of legs be a and b, and the length of the hypotenuse be c. A is an acute angle. With reference to angle A, the ratio of the length of the side adjacent to angle A to length of the hypotenuse is defined as: cosine A = =b/c Cosine A is abbreviated to Cos A. Thus, cos A = b/c. length of side adjacent to angle A length of the hypotenuse hypotenuse leg 1 leg 2 c a b A CLICK TO CONTINUE

38. The tangent of an acute angle... Let the lengths of legs be a and b, and the length of the hypotenuse be c. A is an acute angle. With reference to angle A, the ratio of the length of the side opposite to angle A to length of the side adjacent to angle A is defined as: tangent A = = a/b tangent A is abbreviated Tan A. Thus, tan A = a/b length of side opposite to angle A length of the adjacent side hypotenuse leg 1 leg 2 c a b A CLICK TO CONTINUE

39. SOHCAHTOA This is a clever technique most people use to remember these three basic trig ratios. SOH-CAH-TOA sounds strange? What if I told you it was the ancient oriental queen who loved Geometry? (not true!) S = Sine O = Opposite H = Hypotenuse - C = Cosine A = Adjacent H = Hypotenuse - T = Tangent O = Opposite A = Adjacent CLICK TO CONTINUE

40. Example… sine, cosine and tangent ratios... Find the sine of the given angle. [ SOHCAHTOA ] 53.1 0 20 16 12 B Sin B = Opposite/Hypotenuse sin 53.1 0 = 16/20= 4/5 = 0.80 Cos B= Adjacent/Hypotenuse Cos 53.1 0 = 12/20 = 3/5 = 0.60 Tan B= Opposite/Adjacent Tan 53.1 0 = 16/12 = 5/3 =1.67 CLICK TO CONTINUE

41. Try it yourself … trig ratios... Find the value of sine, cosine, and tangent of the given acute angle. [ SOHCAHTOA ] 53.1 0 15 12 9 B Click to choose your answer from the choices cos 53.1 0 =? A. 5/3B. 3/5C. 4/3D. 3/4E. 4/5F. 5/4 sin 53.1 0 =? A. 5/3B. 3/5C. 4/3D. 3/4E. 4/5F. 5/4 tan 53.1 0 =? A. 5/3B. 3/5C. 4/3D. 3/4E. 4/5F. 5/4 CLICK TO CONTINUE

42. That’s the way! Outstanding ! SOHCAHTOA would be proud of you. You may want to teach others! Click: Go BACK and CONTINUE

43. You can do it… Try again!  I am sorry. Remember it is SOHCAHTOA all the way! You must have used the wrong ratio! Click to Review & Try Again!

44. Angle or Side Lengths? Does the trig ratio depend on the size of the angle or size of the side length? Let us consider similar triangles in our investigation. 5 4 3 36.87 0 A 10 8 6 36.87 0 A 15 12 9 36.87 0 A CLICK TO CONTINUE

45. Find the trig ratios of each... Compute the ratios and make a conjecture 5 4 3 36.87 0 A 10 8 6 36.87 0 A 15 12 9 36.87 0 A sin 36.87 0 ⅗ = 0.6 6/10 = 0.69/15 = 0.6 cos 36.87 0 ⅘ = 0.8 8/10 = 0.812/15 = 0.8 tan 36.87 0 ¾ =0.756/8 = 0.759/12 = 0.75 Conjecture: Trigonometric ratios are a property of similarity (angles) and not of the length of the sides of a right triangle. [Remember SOHCAHTOA!] CLICK TO CONTINUE

46. Computed Trig Ratios... The trig ratios are used so often that technology makes these values readily available in the form of tables and on scientific calculators. We will now show you how to use your calculator to find some trig ratios. Grab a scientific calculator and try it out. CLICK TO CONTINUE

47. How to find trig values on calculator Each calculator brand may work a little differently, but the results will be the same. Look for the trig functions on your calculator: sin, cos and tan select the trig ratio of your choice followed by the angle in degrees and execute (enter). o example: sin 30 will display 0.5 on some calculators you may have to type in the angle first then the ratio o example: 30 sin will display 0.5 CLICK TO CONTINUE

48. Given an angle, use calculator to evaluate trig ratio: Use your calculator to verify that the sine, cosine and tangent of the following angles are correct (to 4 decimals): Angle Asin Acos Atan A 45 o Sin 45 o =0.7071 Cos 45 o =0.7071 Tan 45 o =1.0000 60 o Sin 60 o =0.8660 Cos 60 o =0.5000 Tan 60 o =1.7321 30 o Sin 30 o =0.5000 Cos 30 o =0.8660 Tan 30 o =0.5774 82.5 o Sin 82.5 o =0.9914 Cos 82.5 o =0.1305 Tan 82.5 o =7.5958 CLICK TO CONTINUE

49. Try it yourself... Find the values of the following trig ratios to four decimal places: CLICK TO CONTINUE sin 34 o = ? A. 0.8290B. 0.6745C. 0.5592 cos 56 o = ? A. 0.5592B. 0.8290C. 1.4826 tan 72 o = ? A. 0.3090B. 0.9511C. 3.0777

50. Going backwards: finding angles when we know a trig ratio … We can use the reverse operation of a trig ratio to find the angle with the known trig ratio (n/m) The inverse trig ratios are as follows: Inverse of sin (n/m) is sin -1 (n/m) Inverse of cos (n/m) is cos -1 (n/m) Inverse of tan (n/m) is tan -1 (n/m) CLICK TO CONTINUE

51. Example inverse operation (sin A)  Suppose we know the trig ratio and we want to find the associated angle A. 5 4 A From SOHCAHTOA, we know that from the angle A, we have the opposite side and the hypotenuse. Therefore the SOH part helps us to know that we use sin A = O/H = 4/5 The inverse is thus sin -1 (4/5) = A A = Sin -1 (4/5) = 53.13 o CLICK TO CONTINUE

52. Example inverse operation (cos B)  Suppose we know the trig ratio and we want to find the associated angle B. From SOHCAHTOA, we know that from the angle B, we have the adjacent side and the hypotenuse. Therefore the CAH part helps us to know that we use cos A = A/H = 4/5 The inverse is thus cos -1 (4/5) = B B = cos -1 (4/5) = 36.87 o 5 4 B A CLICK TO CONTINUE

53. Example inverse operation (tan A)  Suppose we know the trig ratio and we want to find the associated angle A. 4 A From SOHCAHTOA, we know that from the angle A, we have the opposite side and the adjacent side. Therefore the TOA part helps us to know that we use tan A = O/A = 4/3 The inverse is thus tan -1 (4/3) = A A = tan -1 (4/3) = 53.13 o 3 CLICK TO CONTINUE

54. Try it yourself…(use inverse) Use a calculator to find the measure of the angles A and B. Use SOHCAHTOA as a guide to what ratio to use. CLICK TO CONTINUE A B C 12 15 19.21 m  A =? A. 38.7B. 51.3C. 53.1A. 38.7B. 51.3C. 53.1 m  B =? A. 38.7B. 51.3C. 53.1A. 38.7B. 51.3C. 53.1

55. Perfect! You make me smile! sin -1 (x) is also referred to as arcsin(x) cos -1 (x) is also referred to as arccos(x) tan -1 (x) is also referred to as arctan(x) Click: Go BACK and CONTINUE

56. Looks like you’re in trouble!  I am sorry. Not to worry, I can help. Click to GO BACK, REVIEW & TRY Again!

57. Finding the legs of a right  Use trig ratios to find sides of a triangle. Remember SOHCAHTOA ! 30 0 A 12 a b With reference to angle A, ●b is the length of side adjacent and ●a is the length of the side opposite the angle. ●the hypotenuse is given Strategy: make an equation that uses only one leg and the hypotenuse at a time. CLICK TO CONTINUE

58. Finding the legs of a right  The tangent ratio may not easily help you figure out the legs a and b in this case. (S OHCAHTOA !) 30 0 A 12 a b Using tangent: tan A = O/A Substituting values from the tgriangle: tan 30 o = a/b From the calculator: tan 30 o = 0.5774 Thus tan 30 o = a/b 0.5774 = a/b And, a = 0.5774(b) GETS YOU STUCK! CLICK TO CONTINUE

59. Finding the legs of a right Using sine ratio to find the leg of a triangle. Remember SOHCAHTOA ! 30 0 A 12 a b Using sine: sin A = O/H Substituting values from the tgriangle: Sin 30 o = a/12 From the calculator: sin 30 o = 0.5 Thus sin 30 o = a/12 0.5 = a/12 And a = 0.5(12) = 6 CLICK TO CONTINUE

60. Finding the legs of a right  Using the cosine ratio to find legs of a triangle. Remember SOHCAHTOA ! 30 0 A 12 a b Using cosine: cos A = A/H Substituting values from the tgriangle: cos 30 o = b/12 From the calculator: cos 30 o = 0.8660 Thus cos 30 o = b/12 0.866 = b/12 And b = 0.866(12) = 10.39 CLICK TO CONTINUE

61. Try it yourself… legs Find the lengths of the legs of the triangle and the third angle. Choose the correct answer. 21 25 o C B A a b CLICK TO CONTINUE m  B = ? A. 90B. 65C. 25 a = ? A. 10.57A. 10.57 B. 22.66C. 21B. 22.66C. 21 b = ? A. 21B. 10.57A. 21B. 10.57 C. 22.66C. 22.66

62. Having trouble?  I am sorry you are having problems. Not to worry, I can help. Click to GO BACK, REVIEW & TRY Again!

63. You got it! Fantastic ! You are on the right track. Click: Go BACK and CONTINUE

64. Finding the hypotenuse...  Use trig ratios to find the hypotenuse of a triangle. Remember SOHCAHTOA ! 30 0 A c 12 b With reference to angle A, ●b is the length of side adjacent and ●12 is the length of the side opposite the angle. ●c is the hypotenuse Strategy: make an equation that uses only one unkown at a time. CLICK TO CONTINUE

65. Finding the hypotenuse...  Use trig ratios to find the hypotenuse of a triangle. Remember SOHCAHTOA ! 30 0 A c 12 b Since 12 is opposite to the angle, we use the sine ratio: Sine A = O/H Substituting values from the tgriangle: sin 30 o = 12/c From the calculator: sin 30 o = 0.5 Thus sin 30 o = 12/c or 0.5 = 12/c c = 12/0.5 = 24 CLICK TO CONTINUE

66. Try it yourself… hypotenuse Find the lengths of the hypotenuse, leg b and the third angle. Choose the best answer. c C B A 13 b 55 o CLICK TO CONTINUE m  A = ? A. 35B. 45C. 55 b = ? A. 22.66A. 22.66 B. 10.57C. 18.57B. 10.57C. 18.57 c = ? A. 18.57A. 18.57 B. 10.57 C. 22.66B. 10.57C. 22.66

67. Terrific! I am proud of your progress! Click: Go BACK and CONTINUE

68. Help is a click away!  Review and try again. PLEASE GO BACK, REVIEW & TRY againPLEASE GO BACK, REVIEW & TRY again.

69. Putting it all together… We now have the tools we need to solve any right triangle (to determine the lengths of each and all sides and the angles, given minimal information) Remember SOHCAHTOA! Typically you get two pieces of information: One side length and one angle or Two sides’ lengths CLICK TO CONTINUE

70. Solve the triangle (side + angle) Given one side length and one angle, determine the rest. Remember SOHCAHTOA ! B 42 0 A 12 C c b Find measure of angle B and side lengths AC and AB. Since we know two angles (90 and 42) we can determine the 3 rd from the Triangle Angle Sum Theorem: m  B = 180 0 –(90 0 +42 0 ) = 48 0. CLICK TO CONTINUE

71. Solve the triangle (side + angle)  Given one side length and one angle, determine the rest. Remember SOHCAHTOA ! 42 0 A 12 B C c b Strategy: side with length 12 is opposite to angle A. To find b, use tan A and to find c, use sin A tan A = O/A tan 42 = 12/b 0.9004 = 12/b b = 12/0.9004 b = 13.33 sin A = O/H sin 42 = 12/c 0.6691 = 12/c c = 12/0.6691 c = 17.93 CLICK TO CONTINUE

72. Try it yourself… side + angle  Solve the triangle. Choose and check answer. 44 0 A 23 B C c b CLICK TO CONTINUE m  B = ? A. 46B. 44C. 23 c = ? A. 33.11B. 23.82C. 23 b = ?A. 23B. 23.82C. 33.11

73. Congratulations! Its fun when you get it! You are on the right track. Click: Go BACK and CONTINUE

74. Sorry, not correct!  I am sorry. Remember it is SOHCAHTOA all the way! You must have used the wrong ratio!  Click to Review & Try Again! Click to Review & Try Again!

75. Solve the triangle (2 sides) Given two side lengths, solve the triangle. Remember SOHCAHTOA ! C A B 17 a 10 Strategy: use Pythagorean Theorem to find the 3 rd side length, a. Use cosine ratio to find measure of angle A Use the Triangle Angle Sum Theorem to find the measure of angle B. CLICK TO CONTINUE

76. Solve the triangle (2 sides)  Given two side lengths, solve the triangle. Remember SOHCAHTOA ! BC A 17 a 10 Using Pythagorean Theorem to find the 3 rd side length, a. c 2 = a 2 + b 2 Pythagorean Theorem 17 2 = a 2 + 10 2 Substituting values 289= a 2 + 100Evaluating the squares a 2 = 289-100Addition property of = a 2 = 189Simplifying a= sqrt(189) = 13.75Taking square root. CLICK TO CONTINUE

77. Solve the triangle (2 sides) Given two side lengths, solve the triangle. Remember SOHCAHTOA ! C A B 17 a 10 Using cosine ratio to find measure of angle A cos A = A/H(the CAH part) cos A = 10/17(substituting values) m  A = cos -1 (10/17)(inverse of cosine) m  A = 53.97 o (Calculator) CLICK TO CONTINUE

78. Try it yourself…(2 sides) Solve the triangle. Click to check your answer… B 45 18 C A c CLICK TO CONTINUE m  A = ? A. 21.8B. 68.2C. 48.5 c = ? A. 21.8B. 68.2C. 48.5 m  B = ? A. 21.8A. 21.8B. 68.2C. 48.568.2C. 48.5

79. You have mastered lot! Congratulations ! We are almost done! You did it again!  Click: Go BACK and CONTINUE Click: Go BACK and CONTINUE

80. Sorry you got it wrong!  I am sorry. Go back and try again. A little review will surely help.  Click to Review & Try Again! Click to Review & Try Again!

81. Real life Applications… Trigonometry is used to solve real life problems. The following slides show a few examples where trigonometry is used. Search the Internet for more examples if you like. CLICK TO CONTINUE

82. Real life example 1 Measuring the height of trees What would you need to know in order to calculate the height of this tree? What trig ratio would you use? Click here to see if we agree. CLICK TO CONTINUE

83. Finding height of towers..  Tall buildings (skyscrapers), towers and mountains… CLICK TO CONTINUE

84. Solve real problem… Assume the line in the middle of the drawn triangle is perpendicular to the beach line. How far is the island from the beach? Click here to check my solution and compare with yours CLICK TO CONTINUE

85. Distance to the beach…(Ans.) The distance we want is the shortest distance “?”. The tangent ratio can be used here: tan 30 o = x/50 0.57735 = x/50 x = 0.57735(50) =28.8675 Therefore the island is about 29m from the beach. Click here to go back

86. Real life example 1 Answer… h is the height of the tree. That is what we are looking for. We need to know the angle of elevation  and also the horizontal distance from A to the bale od the tree, x The tangent ratio would be used: tan  = h/x and so h = x*tan  GO BACK

87. Congratulations Be proud of yourself. You have successfully completed a crash course in basic trigonometry and I expect you to be able to do well on this strand in the Common Core States Standards test. Print the certificate to show your achievement. CLICK TO CONTINUE

88. Certificate of Completion  I hereby certify that _____________________________________ has satisfactorily completed a basic course in Introduction to Trigonometry on this day the ____________________ of the year 20___ The bearer is qualified to solve some real world problems using trigonometry. Signed: Nevermind E. Chigoba