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Trigonometric Ratios

Trigonometric Ratios. Lesson 12.1 HW: 12.1/1-22. Warm – up. Find the missing measures. Write all answers in radical form. x. 30 °. 30 . 10. z. 45 . 3. 60 . y. 45 . 60 °. y. What is a trigonometric ratio?.

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Trigonometric Ratios

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  1. Trigonometric Ratios Lesson 12.1 HW: 12.1/1-22

  2. Warm – up Find the missing measures. Write all answers in radical form. x 30° 30 10 z 45 3 60 y 45 60° y

  3. What is a trigonometric ratio? The relationships between the angles and the sides of a right triangle are expressed in terms of TRIGONOMETRIC RATIOS.

  4. We need to do some housekeeping before we can proceed…

  5. In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE. • Several things MUST BE understood: • This is the hypotenuse.. • This will ALWAYS be the hypotenuse • This is 90°… this makes the right triangle a right triangle….

  6. θand are symbols for unknown angle measures. Their names are ‘Theta’ and ‘Alpha’, from the Greek alphabet. Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized. One more thing…

  7. The 2 other angles and the 2 other sides We will refer to the sides in terms of their proximity to the angle If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. A hypotenuse adjacent opposite

  8. Greek letter ‘PHI’ hypotenuse Adjacent side Opposite side

  9. If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. hypotenuse opposite B adjacent

  10. Opposite side hypotenuse Greek Letter ‘Theta’ Adjacent side

  11. Remember we won’t use the right angle X

  12. Here we go!!!!

  13. The Trigonometric Functions we will be looking at SINE COSINE TANGENT

  14. The Trigonometric Functions SINE COSINE TANGENT

  15. SINE sin Pronounced “sign”

  16. COSINE cos Pronounced “co-sign”

  17. TANGENT tan Pronounced “tan-gent”

  18. The Trigonometric Ratios So, what does this stuff mean?...

  19. hypotenuse hypotenuse opposite opposite adjacent adjacent

  20. We need a way to remember all of these ratios…

  21. Some Old Hippie Came A Hoppin’ Through Our Old Hippie Apartment

  22. Sin Opp Hyp Cos Adj Hyp Tan Opp Adj Old Hippie

  23. SOHCAHTOA

  24. For any right-angled triangle Definition of Sine Ratio opposite side sin = hypotenuse

  25. For any right-angled triangle Definition of Cosine Ratio adjacent side cos  = hypotenuse

  26. For any right-angled triangle Definition of Tangent Ratio. opposite side tan  = adjacent

  27. Find: Sin 16° Tan 58°

  28. Ex. 1: Finding Trig Ratios opposite 8 4 sin A = ≈ 0.4706 hypotenuse 0.4706 ≈ 17 8.5 SohCahToa 15 7.5 adjacent ≈ 0.8824 ≈ cos A = 0.8824 hypotenuse 8.5 17 opposite 8 4 0.5333 tan A = 0.5333 ≈ ≈ adjacent 7.5 15 Trig ratios are often expressed as decimal approximations.

  29. Ex. 2: Finding Trig Ratios SohCahToa opp 5 sin S = 0.3846 ≈ hyp 13 adj 12 ≈ 0.9231 cos S = 13 hyp opp 5 ≈ 0.4167 tan S = adj 12

  30. Finding sin, cos, and tan.(Just writing a ratio or decimal.)

  31. Find the sine, the cosine, and the tangent of angle A. Give a fraction and decimal answer (round to 4 places). 10.8 9 A 6 Shrink yourself down and stand where the angle is. Now, figure out your ratios.

  32. Find the sine, the cosine, and the tangent of angle A 24.5 Give a fraction and decimal answer (round to 4 decimal places). 8.2 A 23.1 Shrink yourself down and stand where the angle is. Now, figure out your ratios.

  33. Finding an angle.(Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons.)

  34. 17.2 9 ) 2nd tan Ex. 1: Find . Round to four decimal places. 17.2 9 Shrink yourself down and stand where the angle is. SohCahToa Now, figure out which trig ratio you have and set up the problem. Make sure you are in degree mode (not radians).

  35. 7 23 ) 2nd cos Ex. 2: Find . Round to three decimal places. 7 23 SohCahToa Make sure you are in degree mode (not radians).

  36. Finding the angle measure  4 7 In the figure, find  opposite side sin  = hypotenuse 4 sin  = 7  = 34.85

  37. Finding the length of a side In the figure, find y opposite side y sin 35 = hypotenuse y sin 35 = 35° 11 11 11* sin35 = y y = 6.31

  38. Finding the angle measure In the figure, find  adjacent Side 3 cos  =  hypotenuse 3 = cos  = 8 8  = 67.98

  39. Finding the length of a side In the figure, find x adjacent side 6 cos 42 = hypotenuse 42° 6 cos 42 = x x 6 x = cos 42 x = 8.07

  40. Finding the angle measure In the figure, find  3 opposite side tan  = adjacent side 3 tan  = 5 = 5   = 78.69

  41. Finding the length of a side In the figure, find z z Opposite side 22 tan 22 = adjacent Side 5 5 tan 22 = z 5 z = tan 22 z = 12.38

  42. Make Sure that the triangle is right-angled Conclusion SohCahToa

  43. Solving a Problem withthe Tangent Ratio SohCahToa We know the angle and the side adjacent to 60º. We want to know the opposite side. Use the tangent ratio: h = ? 60º 53 ft

  44. Note: • The value of a trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

  45. Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30 opposite 1 sin 30= = 0.5 hypotenuse 2 adjacent √3 cos 30= ≈ 0.8660 hypotenuse 2 opposite 1 √3 tan 30= = adjacent ≈ 0.5774 √3 3 Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. 30 √3

  46. Ex: 5 Using a Calculator • You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

  47. Sample keystrokes sin sin ENTER 74 74 COS COS ENTER 74 or 74 TAN TAN ENTER

  48. Notes: • If you look back at Examples 1-5, you will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.

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