Chapter 11 Trigonometric Ratios 11.2 The Sine Ratio
Introduction The figure below shows a right-angled triangle ABC, where B = and C = 90. A hypotenuse B θ adjacent side of opposite side of C AB is called the hypotenuse; BC is called the adjacent side of ; AC is called the opposite side of.
Worksheet 1 Right-angled triangle with a given acute angle 30 o Part 1 Measure the length of the opposite sides and the hypotenuses by a ruler for the following right-angled triangles with a given acute angle 30 o of different size. Calculate their ratio and complete the following table.
Part 1 Worksheet 1 Right-angled triangle with a given acute angle 30 o 1.5 3 2.5 0.5 5 1.4 0. 2.8 2 0. 4 1.8 0. 3.6 1.5 3 0. 5 2.5 5 1.4 2.8 5 2 4 5 1.8 3.6 5
0.5 0.5 1 A 2 30 2 B 4 30 3 C 6 30 For a right-angled triangle with a given acute angle 30 o, opposite side is equal to 0.5. hypotenuse
0.5 Conclusion: For a right-angled triangle with a given acute angle, opposite side is a constant. hypotenuse
Worksheet 2 The Ratio of the opposite side to the hypotenuse for different acute angles Part 1 A right-angled triangle will be shown on the screen. We drag the red point on the right hand side to change the size of the right-angled triangle. And then we drag the green point on the left hand side to change the angles of the right-angled triangle. Drag the red point to change the size of the right-angled triangle. Drag the green point to change the angles of the right-angled triangle. Opposite side = 0.5 Hypotenuse = 1 The ratio = 0.5 http://geogebraweb.appspot.com/app.html
Drag the red point to change the size of the right-angled triangle. Drag the green point to change the angles of the right-angled triangle. Opposite side = 0.5 Hypotenuse = 1 The ratio = 0.5 0.09 0.17 0.26 0.34 0.42 0.5 0.57 0.64 0.71 0.77 0.82 0.87 0.91 0.94 0.97 0.98 increase 0 1
x 4 0.5 x = 4 0.5 = 2 1. 2. x 5 0.34 10 0.71 x 0.71 x = 10 x 10 0.71 x = 14.1 x = 5 0.34 x = 1.7
3. 4. x 6 0.87 x = 6 0.87 x = 5.2 6 0.85 = 45 o 0.71 5. 6. 10 0.77 13 = 50 o 12.5 0. 98 12.7 = 80 o
Let us think 1. 2.
In the figure, the sine of angle is denoted by sin, and opposite side sin =. hypotenuse A trigonometric ratio such as sin is a ratio and therefore has no unit.
Find the value of sin in the figure. sin = = = opposite side (YZ) hypotenuse (XY) 2 10 1 5
Question 1 In the following figures, find sin θ. (Give your answer in 3 significant figures if necessary.) (a) 15 8 (b) 12 17 θ θ 20 16 Solution (a) sin θ 15 17 = 0.882 (b) sin θ 16 20 4 5 = 0.8
Question 2 Find the values of the following. (a) sin Q (b) sin R Give your answer in fraction. Solution (a) sin Q PR QR (b) sin R PQ QR 8 10 4 5 6 10 3 5
Worksheet 2 Calculator and the Sine Ratio Part 1 By using a calculator, find the following value. angle (2 d.p.) 5 11.54 o 20 23 36.87 o 53.13 o 61 64.16 o 71.81 o 82 sin (4 d.p.) 0.0872 0.2 0.3420 0.3907 0.6 0.8 0.8746 0.9 0.95 0.9903
Part 2 Do the following questions on your C.W. book. Question 1 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) (b) sin 43 sin 28 = 0.2125 (cor. to 4 d.p.) 2 sin 11 = 0.3816 (cor. to 4 d.p.) sin 43 = 0.681 99, sin 28 = 0.469 47 sin 11 = 0.190 80
Part 2 Do the following questions on your C.W. book. Question 2 By using a calculator, find the values of the following expressions correct to 4 decimal places. (a) sin 66 (b) sin 32.48 Solution (a) Keying sequence Display sin 66 EXE 0.913545457 sin 66 0.9135 (cor. to 4 d.p.) (b) Keying sequence Display sin 32.48 EXE 0.537005176 sin 32.48 0.5370 (cor. to 4 d.p.)
Part 2 Do the following questions on your C.W. book. Question 3 (a) (b) By using a calculator, find the value of sin 34 + sin 26 sin 60 correct to 3 significant figures. From the result obtained in (a), is sin 34 + sin 26 equal to sin (34 + 26)? Solution (a) Keying sequence Display sin 34 + sin 26 sin 60 EXE 0.131538646 (b) sin 34 sin 26 sin 60 0.132 (cor. to 3 sig. fig.) sin 34 sin 26 sin 60 sin 34 sin 26 sin 34 sin 26 0 sin 60 sin (34 26)
Part 2 Do the following questions on your C.W. book. Question 4 Find the acute angle in each of the following using a calculator. (Give your answers correct to 3 significant figures.) (a) sin = 0.22 (b) sin = sin 68 sin 40 Solution (a) sin = 0.22 = 12.7 (cor. to 3 sig. fig.) (b) sin = sin 68 sin 40 = 0.2844 = 16.5 (cor. to 3 sig. fig.) sin 68 = 0.927 18, sin 40 = 0.642 78
Part 2 Do the following questions on your C.W. book. Question 5 Find the acute angles in the following using a calculator. (a) sin 0.62, correct to the nearest degree. (b) 1 sin sin 35, correct to the nearest 0.1. 5 (c) 7 sin 3, correct to 3 significant figures. Solution (a) Keying sequence Display SHIFT sin 0.62 EXE 38.31613447 sin 0.62 38 (cor. to the nearest degree)
Solution (b) Keying sequence Display (c) SHIFT sin ( 1 5 sin 35 ) EXE 6.587203533 1 sin sin 5 6.6 7sin sin 3 3 7 35 (cor. to the nearest 0.1) Keying sequence Display SHIFT sin ( 3 7 ) EXE 25.37693352 sin 3 7 25.4 (cor. to 3 sig. fig.)
Part 2 Do the following questions on your C.W. book. 1.27 1.38 3.37 7.59 1.125 As sin90 o =11.125 It is not true. = 25.6 o As (10+15) o = 25 o is not equal to (10+15) o.
Part 3 Use a calculator to find the unknown in each of the following right-angled triangles. (Give the answer correct to 3 significant figures.) x 15 sin 43 o x = 15 sin43 o x = 10.2 1. 2. x 3 sin 40 o y 10.5 o sin 78 y = 10.5 sin 78 o x = 3 sin 40 o x = 1.7 y = 10.3
7 sin 25 y 3. 4. y o 7 o sin 25 sin 5 8 = 38.7 o y = 16.6 5. sin 0.7 1.3 = 32.6 o
Part 4 Do Ex.11A Q.13-18 (P.180) on your C.W. book. = 16 o = 53 o = 67 o x = 7.07 x = 22.7 x = 3.76
Part 4 Do Ex.11A Q.13-18 (P.180) on your C.W. book. sin 7 25 = 16 o sin 8 10 = 53 o sin 36 39 12 13 = 67 o
Part 4 Do Ex.11A Q.13-18 (P.180) on your C.W. book. x 10 sin 45 o x = 10 sin 45 o x = 7.07 9.6 x x o sin 25 9.6 o sin 25 o 5 sin 36.5 x = 22.7 x = 3.76 x x 5 o sin 36.5
Conclusion: 1. In the figure, the sine of angle is denoted by sin, and sin = opposite side hypotenuse 2. For a right-angled triangle with a given acute angle, a constant the sine ratio of is. increase 3. (a) When increases, the sine ratio of will. 0 1 (b) When 0 o < < 90 o, the sine ratio of lies between and. 4. The sine ratio of any acute angle can be easily found by using a calculator. We should make sure that the calculator is set in degree mode before doing the calculation.