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Sine Ratio

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Sine Ratio

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  1. Sine Ratio

  2. A C B Introduction to Trigonometric Ratios The figure below shows a right-angled triangle ABC, where B =  and C = 90. hypotenuse opposite side of  adjacent side of  AB is called the hypotenuse; BC is called the adjacent sideof  ; AC is called the opposite sideof  .

  3. In fact, the size of  has certain relationship between the ratios Consider the right-angled PQRbelow. Is there any relationship among, PQ, QR and PR? and PQ P PR . , PQ QR R Q QR PR These ratiosare known as trigonometric ratios. I only know that PQ2 + QR2 = PR2… How does the size of  relate to the sides of the triangle?

  4. 6 4 3 2 2 C B 1 A 30 30 30 opposite side Triangle hypotenuse Complete the table below. What do you observe? For a right-angled triangle with a given acute angle , = 2 1 1 1 3 opposite side 4 6 2 2 2 is a constant. hypotenuse = Consider the following three right-angled triangles.

  5. Concept of Sine Ratio The sine ratio of an acute angle  is defined as below: hypotenuse opposite side of 

  6. 30 30 30 sin 30 = = = 1 2 3 6 2 4 For a right-angled triangle with a given acute angle , the sine ratio of is a constant. For example, 6 3 4 2 2 1

  7. A 13 5 B C  AC is the opposite side of B, and AB is the hypotenuse.

  8. Follow-up question 1 (a) 12 (b) 16 17 15 20 8 Solution

  9. Example 1 Solution

  10. Make sure that the calculator is set in degree mode. Degree mode is usually denoted by the key DEG or D on calculators. 2. Use the key sin on a calculator to find the value of sin . sin 30 EXE Finding Sine Ratio Using Calculators Find sin  for a given angle  For example, the value of sin 30 can be obtained by keying: The answer is 0.5.

  11. = 0.2125 (cor. to 4 d.p.) = 0.3816 (cor. to 4 d.p.) Follow-up question 2 By using a calculator, find the values of the following expressions correct to 4 significant figures. (a) sin 43 – sin 28 (b) 2 sin 11 Solution (a) sin 43 – sin 28  sin 43 = 0.681 99…, sin 28 = 0.469 47… (b) 2 sin 11  sin 11 = 0.190 80…

  12. Example 2 Solution

  13. Example 3 Solution

  14. In degree mode, use the keys SHIFT and sin to find the corresponding acute angle . SHIFT sin 0.5 EXE Find  for a given value of sin  For example, given that sin  = 0.5, can be obtained by keying: The answer is 30, i.e.  = 30.

  15.  = 12.7 (cor. to 3 sig. fig.)  = 16.5 (cor. to 3 sig. fig.) Follow-up question 3 Find the acute angle  in each of the following using a calculator. (Give your answers correct to 3 significant figures.) • sin  = 0.22 (b) sin  = sin 68 – sin 40 Solution • sin  = 0.22 (b) sin  = sin 68 – sin 40  sin 68 = 0.927 18…, sin 40 = 0.642 78… = 0.2844…

  16. Example 4

  17. Solution

  18. In ABC, C = 90, B = 55and AB = 8 m. A 8 m B C Using Sine Ratio to Find Unknowns in Right-Angled Triangles We can use the sine ratio to solve problems involving right-angled triangles. Find AC correct to 2 decimal places.

  19. P In PQR, R = 90, PQ = 9 m and PR = 7 m. 7 m 9 m R Q Find Q correct to 2 decimal places.

  20. C 7 cm B A Follow-up question 4 Find AB correct to 2 decimal places. Solution

  21. Example 5 Solution

  22. Example 6 Solution

  23. Example 7 Solution

  24. Cosine Ratio

  25. Concept of Cosine Ratio The cosine ratio of an acute angle  is defined as below: hypotenuse adjacent side of 

  26. 60 60 60 cos 60 = = = 1 2 3 4 2 6 For a right-angled triangle with a given acute angle , the cosine ratio of  is a constant. For example, 6 4 2 1 3 2

  27. In ABC, C = 90, AB = 5.2 and AC = 2. A 5.2 2 B C  AC is the adjacent side of A, and AB is the hypotenuse.

  28. 6 3 5 8 10 4 Follow-up question 5 (a) (b) Solution

  29. Example 8 Solution

  30. In degree mode, use the key cos to find the value of cos . cos 30 EXE Finding Cosine Ratio Using Calculators Find cos  for a given angle  For example, the value of cos 30 can be obtained by keying: The answer is 0.8660…

  31. By using a calculator, find the values of the following expressions correct to 3 significant figures. = 4.37 (cor. to 3 sig. fig.) (b) = 0.512 (cor. to 3 sig. fig.) Follow-up question 6 Solution (a) 5 cos 29  cos 29 = 0.874 61…  cos 40 = 0.766 04…, cos 75 = 0.258 81…

  32. Example 9 Solution

  33. In degree mode, use the keys SHIFT and cos to find the corresponding acute angle . SHIFT cos 0.5 EXE Find  from a given value of cos  For example, given that cos  = 0.5, can be obtained by keying The answer is 60, i.e.  = 60.

  34. Follow-up question 7 Find the acute angle  in each of the following using a calculator. (Give your answers correct to 3 significant figures.) Solution  cos 24 = 0.913 54…

  35. Example 10

  36. Solution

  37. In ABC, C = 90, B = 55 and AB = 8 m. A 8 m B C Using Cosine Ratio to Find Unknowns in Right-Angled Triangles We can use the cosine ratio to solve problems involving right-angled triangles. Find BC correct to 2 decimal places.

  38. P In PQR, R = 90, PQ = 9 m and QR = 7 m. 9 m Q R 7 m Find Q correct to 2 decimal places.

  39. B 3.5 cm C 4 cm A Follow-up question 8 Find B correct to 2 decimal places. Solution

  40. Example 11 Example 12 Example 13

  41. Example 11 Solution

  42. Example 12 Solution

  43. Example 13 Solution

  44. Tangent Ratio

  45. Concept of Tangent Ratio The tangent ratio of an acute angle  is defined as below: opposite side of  adjacent side of 

  46. 45 45 45 tan 45 = = = 1 3 2 2 1 3 For a right-angled triangle with a given acute angle , the tangent ratio of  is a constant. For example, 3 2 1 1 2 3

  47. In ABC, C = 90, AC = 2.4 and BC = 3.2. A 2.4 B C 3.2  AC is the adjacent side of A, and BC is the opposite side of A.