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Chapter 5 Introduction to Trigonometry: 5.7 The Primary Trigonometric Ratios

Chapter 5 Introduction to Trigonometry: 5.7 The Primary Trigonometric Ratios. Humour Break. 5.7 The Primary Trigonometric Ratios . Goals for Today: Learn how to identify sides and angles in triangles Learn the primary trigonometric ratios

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Chapter 5 Introduction to Trigonometry: 5.7 The Primary Trigonometric Ratios

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  1. Chapter 5 Introduction to Trigonometry: 5.7 The Primary Trigonometric Ratios

  2. Humour Break

  3. 5.7 The Primary Trigonometric Ratios Goals for Today: • Learn how to identify sides and angles in triangles • Learn the primary trigonometric ratios • Use the primary trigonometric ratios to find a missing side • Use the primary trigonometric ratios to find a missing angle

  4. 5.7 Primary Trigonometric Ratios In previous math course, you learned how to calculate a missing side in a right angle triangle when you were given two other sides. You did this using the pythagorean theorem a² + b² = c²

  5. 5.7 Primary Trigonometric Ratios You also learned to how to calculate missing angles using angle rules, such as the 180° triangle rule, where the sum of angles in a triangle add up to 180 °

  6. 5.7 Primary Trigonometric Ratios Now, we are going to introduce the trigonometric ratios, where we work with both angles and sides to find unknown angles in sides. Today, we are working with the primary trig ratios which are used for right angle triangles only

  7. 5.7 Primary Trigonometric Ratios What are the sides from the perspective of angle A?

  8. 5.7 Primary Trigonometric Ratios From the perspective of angle A.... Hypotenuse (or side “c”) Opposite (or side “a”) Adjacent (or side “b”)

  9. 5.7 Primary Trigonometric Ratios What are the sides from the perspective of angle B?

  10. 5.7 Primary Trigonometric Ratios From the perspective of angle B.... Hypotenuse (or side “c”) Adjacent* (or side “a”) Opposite* (or side “b”) * Side name changes because from the perspective of a different triangle

  11. 5.7 Primary Trigonometric Ratios For a right angle triangle, the three primary trig ratios are:

  12. 5.7 Primary Trigonometric Ratios An acronym to help remember these formulas is SOHCAHTOA

  13. 5.7 Primary Trigonometric Ratios With these ratios you are dealing with one of two situations • You have two sides in a triangle and you use them to find an angle by using the inverse or 2nd of SIN, COS, or TAN on your calculator • You have a side and an angle in a triangle and you want to find another side by using the SIN, COS, or TAN on your calculator

  14. 5.7 Primary Trigonometric Ratios Consider the classic 3, 4, 5 right triangle

  15. 5.7 Primary Trigonometric Ratios Let’s use scenario 1 to find angle A. Which ratio?

  16. 5.7 Primary Trigonometric Ratios SIN because side a = 3 (opposite) and side c = 5 (hypotenuse) from angle A

  17. 5.7 Primary Trigonometric Ratios SIN ∠ A = opposite hypotenuse Hint: Once you have the trig ratio, you use your SIN to -1 or inverse function to convert the ratio 0.6 to the angle of 36.9°

  18. 5.7 Primary Trigonometric Ratios Your calculator does a nice job. In the old days (when I was in high school) you had to look up the ratio in a table and convert it into an angle!

  19. 5.7 Primary Trigonometric Ratios Let’s again use scenario 1 to find angle A. Which ratio? Why did it change?

  20. 5.7 Primary Trigonometric Ratios COS because side b = 4 (adjacent) and side c = 5 (hypotenuse) from angle A

  21. 5.7 Primary Trigonometric Ratios COS ∠ A = adjacent hypotenuse Hint: Once you have the trig ratio, you use your COS to -1 or inverse function to convert the ratio 0.8 to the angle of 36.9° We got the same angle which makes sense!... Same triangle!

  22. 5.7 Primary Trigonometric Ratios Finally, let’s again use scenario 1 to find angle A. Which ratio?

  23. 5.7 Primary Trigonometric Ratios TAN because side b = 4 (adjacent) and side a = 3 (opposite) from angle A

  24. 5.7 Primary Trigonometric Ratios TAN∠ A = opposite adjacent Hint: Once you have the trig ratio, you use your TAN to -1 or inverse function to convert the ratio 0.75 to the angle of 36.9° We got the same angle which makes sense!... Same triangle!

  25. 5.7 Primary Trigonometric Ratios Let’s use scenario 1 to find angle B. Which ratio? What’s different than when we were finding angle A?

  26. 5.7 Primary Trigonometric Ratios When we were finding angle A, given these sides, we used the SIN ratio. From angle B we are dealing with the adjacent sides and the hypotenuse so we have to use the COS ratio.

  27. 5.7 Primary Trigonometric Ratios COS ∠ B = adjacent hypotenuse Hint: Once you have the trig ratio, you use your COS to -1 or inverse function to convert the ratio 0.6 to the angle of 53.1°

  28. 5.7 Primary Trigonometric Ratios Let’s use scenario 1 to find angle B. Which ratio? What’s different than when we were finding angle A?

  29. 5.7 Primary Trigonometric Ratios When we were finding angle A, given these sides, we used the COS ratio. From angle B we are dealing with the opposite sides and the hypotenuse so we have to use the SIN ratio.

  30. 5.7 Primary Trigonometric Ratios SIN ∠ B = opposite hypotenuse Hint: Once you have the trig ratio, you use your SIN to -1 or inverse function to convert the ratio 0.6 to the angle of 36.9°

  31. 5.7 Primary Trigonometric Ratios Let’s use scenario 1 to find angle B. Which ratio? What’s different than when we were finding angle A?

  32. 5.7 Primary Trigonometric Ratios When we were finding angle A, given these sides, we used the TAN ratio. From angle B we are dealing with the opposite sides and the adjacent sides again, so we still use the TAN ratio, but the numbers are reversed

  33. 5.7 Primary Trigonometric Ratios TAN∠ A = opposite adjacent Hint: Once you have the trig ratio, you use your TAN to -1 or inverse function to convert the ratio 1.3333 to the angle of 53.1° We got the same angle which makes sense!... Same triangle!

  34. 5.7 Primary Trigonometric Ratios Now, lets again consider the classic 3, 4, 5 right triangle, but this time, given an angle and a side and asked to find a side

  35. 5.7 Primary Trigonometric Ratios From the perspective of angle A, we are dealing with the opposite side and the hypotenuse… so we have to use the SIN ratio…

  36. 5.7 Primary Trigonometric Ratios Now, lets again consider the classic 3, 4, 5 right triangle, but this time, given angle A and side c and asked to find a side a… This is scenario 2… given an angle and a side… find another side…

  37. 5.7 Primary Trigonometric Ratios SIN ∠ A = opposite hypotenuse You input 36.9 into your calculator and hit the SIN button to get the ratio 0.6004. In some calculators, the order is reversed…

  38. 5.7 Primary Trigonometric Ratios Similiarly, if asked to side side b, from the perspective of angle A, we are here dealing with the adjacent side side and the hypotenuse… which ratio would we use?

  39. 5.7 Primary Trigonometric Ratios The COS ratio because from the perspective of angle A, we are dealing with the adjacent side and the hypotenuse…

  40. 5.7 Primary Trigonometric Ratios COS ∠ A = adjacent hypotenuse You input 36.9 into your calculator and hit the COS button to get the ratio 0.7997. In some calculators, the order is reversed…

  41. 5.7 Primary Trigonometric Ratios Similiarly, if asked to find side b, from the perspective of angle A, but we were given side a… we sould be dealing with the opposite side and the adjacent side side… which ratio would we use?

  42. 5.7 Primary Trigonometric Ratios The Tan ratio, because we are dealing with the opposite and adjacent sides…

  43. 5.7 Primary Trigonometric Ratios Tan ∠ A = opposite adjacent You input 36.9 into your calculator and hit the TAN button to get the ratio 0.7508. In some calculators, the order is reversed…

  44. 5.7 Primary Trigonometric Ratios If we now examine things from the perspective of angle B, we are dealing with the adjacent side and the hypotenuse… so we have to use the COS ratio… We are again dealing with Scenario 2… given a side and an angle, finding another side…

  45. 5.7 Primary Trigonometric Ratios COS ∠ B = adjacent hypotenuse You input 53.1 into your calculator and hit the COS button to get the ratio 0.6004. In some calculators, the order is reversed… Note that COS 53.1° is the same as SIN 36.9°

  46. 5.7 Primary Trigonometric Ratios In this next example… again from the perspective of angle B, we are dealing with the opposite side and the hypotenuse… so we have to use the SIN ratio… We are again dealing with Scenario 2… given a side and an angle, finding another side…

  47. 5.7 Primary Trigonometric Ratios SIN ∠ B = opposite hypotenuse You input 53.1 into your calculator and hit the SIN button to get the ratio 0.7997. In some calculators, the order is reversed…

  48. 5.7 Primary Trigonometric Ratios In this final example… again from the perspective of angle B, we are dealing with the opposite side and the adjacent side… so we have to use the TAN ratio… We are again dealing with Scenario 2… given a side and an angle, finding another side…

  49. 5.7 Primary Trigonometric Ratios TAN ∠ B = opposite adjacent You input 53.1 into your calculator and hit the SIN button to get the ratio 0.7997. In some calculators, the order is reversed…

  50. Homework • Tuesday, December 3rd – p.496, #1, 2, 4, 9-11 • Thursday, December 12th– p.498, #12-15, 17, 19-21 • Friday, December 13th – p.498, #22-26

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