l1. trigonometric function
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This document discusses trigonometric ratios and their use in solving for missing sides and angles of right triangles. It provides examples of using trigonometric ratios like sine, cosine, and tangent to find values of angles and sides when given information about two sides of a right triangle. It also includes practice problems where the learner must use trigonometric functions on a calculator to determine values. Overall, the document aims to teach learners how to apply the six trigonometric ratios to solve for unknown values in right triangles.
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- 2. SOH CAH TOA
- 4. Length of Sides (in cm) Trigonometric Ratios Angle (°) Opposite Adjacent Hypotenuse 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 a. b. c. d. e. f.
- 5. Length of Sides (in cm) Trigonometric Ratios Angle (°) Opposite Adjacent Hypotenuse 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 a. 14° 2cm 7.8cm 8.2cm 0.24 0.97 0.25 b. 24° 3.3cm 5.9cm 7.4cm 0.41 0.91 0.45 c. 54° 6.2cm 5.5cm 7.3cm 0.81 0.59 1.38 d. 65° 5.1cm 2.4cm 5.7cm 0.91 0.42 2.14 e. 30° 3.5cm 4.9cm 6.7cm 0.5 0.87 0.58 f.51° 5.6cm 4.7cm 7.2cm 0.78 0.63 1.23
- 9. Example 1: Triangle BCA is a right- angle at C. If 𝑐 = 23 and 𝑏 = 17, find angle A, angle B and a. Express your answers up to two decimal places.
- 10. cos 𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos 𝜃 = 𝑏 𝑐 cos 𝐴 = 17 23 𝑐𝑜𝑠 𝐴 = 0.7391 𝐴 = cos−1 (0.7391) 𝐴 = 42.34°
- 11. b. Since in part (a), it was already found that ∠𝐴 = 42.34°, Then ∠𝐵 = 90° − 42.34° ∠𝐵 = 47.66°
- 12. c. Using the Pythagorean Theorem: 𝑎2 + 𝑏2 = 𝑐2 𝑎2 + 17 2 = 23 2 𝑎2 + 289 = 529 𝑎2 = 529 – 289 𝑎2 = 240 𝑎 = 240 𝑎 = 15.49
- 13. Example 2: Triangle BCA is right-angle at C if 𝑐 = 27 and ∠𝐴 = 58°, find ∠𝐵, b and a. Solution: a. To find B, since ∠𝐵 𝑎𝑛𝑑 ∠𝐴 are complementary angles, then
- 14. 58° ∠𝐵 + ∠𝐴 = 90° ∠𝐵 = 90° − 58° ∠ 𝐵 = 32°
- 15. b. To find b, since b is the adjacent side of ∠A and c is the hypotenuse of right ΔBCA then use CAH.
- 16. cos 𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cos 𝜃 = 𝑏 𝑐 cos 58° = 𝑏 27 𝑏 = 27 cos 58° 𝑏 = 27(0.5299) =14.31
- 17. c. To find a, since a is the opposite side of ∠A then use________? sin 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑠𝑖𝑛 𝐴 = 𝑎 𝑐 sin 58° = 𝑎 27 𝑎 = 27 𝑠𝑖𝑛 58° 𝑎 = 27 0.8480 𝑎 = 22.90 SOH.
- 18. Ready?
- 19. 1. Triangle ACB is right-angled at C. if ∠𝐴 = 63° and 𝑎 = 11 𝑐𝑚, find ∠B, b and c. 2. Triangle ACB is right-angled at C. If 𝑎 = 18.5 𝑐𝑚 and 𝑏 = 14.2 𝑐𝑚, find c, ∠A and ∠B.
- 20. Solution in #1
- 21. a. To find B, take note that B and A are complementary angles, then ∠𝐵 + ∠𝐴 = 90° ∠𝐵 = 90 °– 63° ∠𝐵 = 27°
- 22. b. To find b, since b is the adjacent side and a is the opposite side of ∠𝐴, then use TOA. 𝑡𝑎𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan 𝐴 = 𝑎 𝑏 tan 63° = 11 𝑏 𝑏 tan 63° = 11 𝑏 1.9626 = 11 𝑏 = 11 1.9626 𝑏 = 5.60𝑐𝑚 b 27°
- 23. c. To find c, since c is the hypotenuse and a is the is opposite side of ∠𝐴, then use SOH. 𝑠𝑖𝑛 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑠𝑖𝑛 𝐴 = 𝑎 𝑐 sin 63° = 𝑎 27 𝑎 = 27 sin 63° 𝑎 = 27 (0.8910) 𝑎 = 24.0571 or 24.06
- 24. Solution in #2
- 25. a. To find c, use Pythagorean theorem 𝑐2 = 𝑎2 + 𝑏2 𝑐2 = 18.5 2 + 14.2 2 𝑐2 = 342.25 + 201.64 𝑐2 = 543.89 𝑐 = 543.89 𝑐 = 23.32
- 26. b. To find ∠A, since a and b are opposite and adjacent side of ∠A respectively, then use TOA. 𝑡𝑎𝑛 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑎𝑛 𝐴 = 𝑎 𝑏 𝑡𝑎𝑛 𝐴 = 18.5 14.2 𝑡𝑎𝑛 𝐴 = 1.3028 𝐴 = tan−1 (1.3028) 𝐴 = 52°
- 27. c. Based on the fact that ∠A and ∠B are complementary, the measure of angle ∠B is 90° – 52° = 38°.
- 28. This lesson was about six trigonometric ratios. Various activities were provided to help students illustrate and define the six trigonometric ratios. They have also learned how to use these concepts in finding the missing sides and angles of a right triangle and applied them to real life situations. Their knowledge in this lesson will help them understand the next topic, which is, the trigonometric ratios of special angles.
- 29. (Things to Remember) The sin and 𝜽 are married to each other. They are not treated as product of sin and 𝜽. They are inseparable.𝒔𝒊𝒏 𝜽 ≠ 𝒔𝒊𝒏 × 𝜽 This is TRUE for all trigonometric ratios.
- 30. Exercise #4.__Directions: Determine the other side, given that two sides are: 1. 𝑎 = 5𝑓𝑡., 𝑏 = 10𝑓𝑡., 𝑐 =? 2. 𝑎 = 4𝑘𝑚, 𝑏 =? 𝑐 = 12𝑘𝑚 3. 𝑎 =?, 𝑏 = 8𝑚, 𝑐 = 24𝑚 4. 𝑎 =?, 𝑏 = 6𝑖𝑛𝑐ℎ, 𝑐 = 9𝑖𝑛𝑐ℎ
- 32. 1.c = 11.18𝑓𝑡 2.b= 11.31𝑘𝑚 3.a= 22.63𝑚 4.a= 6.71𝑖𝑛𝑐ℎ
- 33. Exercise #4.__ Using the Calculator to find Trigonometric Ratios Find the value of the following, correct to two decimal places. a. 𝑐𝑜𝑠 23° b.𝑠𝑖𝑛 65° c.𝑡𝑎𝑛 35° d. 𝑐𝑜𝑠 7° e. 𝑡𝑎𝑛 85°