Chapter 11 Trigonometric Ratios The Sine Ratio

Transcription

1 Chapter 11 Trigonometric Ratios 11.2 The Sine Ratio

2 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.

3 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.

4 Part 1 Worksheet 1 Right-angled triangle with a given acute angle 30 o

5 A B C 6 30 For a right-angled triangle with a given acute angle 30 o, opposite side is equal to 0.5. hypotenuse

6 0.5 Conclusion: For a right-angled triangle with a given acute angle, opposite side is a constant. hypotenuse

7 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

8 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 = increase 0 1

9

10 x x = = x x 0.71 x = 10 x x = 14.1 x = x = 1.7

11 3. 4. x x = x = = 45 o = 50 o = 80 o

12

13 Let us think 1. 2.

14 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.

15 Find the value of sin in the figure. sin = = = opposite side (YZ) hypotenuse (XY)

16 Question 1 In the following figures, find sin θ. (Give your answer in 3 significant figures if necessary.) (a) 15 8 (b) θ θ Solution (a) sin θ = (b) sin θ = 0.8

17 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

18 Worksheet 2 Calculator and the Sine Ratio Part 1 By using a calculator, find the following value. angle (2 d.p.) o o o o o 82 sin (4 d.p.)

19 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 = (cor. to 4 d.p.) 2 sin 11 = (cor. to 4 d.p.) sin 43 = , sin 28 = sin 11 =

20 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 Solution (a) Keying sequence Display sin 66 EXE sin (cor. to 4 d.p.) (b) Keying sequence Display sin EXE sin (cor. to 4 d.p.)

21 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 ( )? Solution (a) Keying sequence Display sin 34 + sin 26 sin 60 EXE (b) sin 34 sin 26 sin (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)

22 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 = = 16.5 (cor. to 3 sig. fig.) sin 68 = , sin 40 =

23 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 (c) 7 sin 3, correct to 3 significant figures. Solution (a) Keying sequence Display SHIFT sin 0.62 EXE sin (cor. to the nearest degree)

24 Solution (b) Keying sequence Display (c) SHIFT sin ( 1 5 sin 35 ) EXE sin sin sin sin (cor. to the nearest 0.1) Keying sequence Display SHIFT sin ( 3 7 ) EXE sin (cor. to 3 sig. fig.)

25 Part 2 Do the following questions on your C.W. book As sin90 o = It is not true. = 25.6 o As (10+15) o = 25 o is not equal to (10+15) o.

26 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 = 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

27 7 sin 25 y y o 7 o sin 25 sin 5 8 = 38.7 o y = sin = 32.6 o

28 Part 4 Do Ex.11A Q (P.180) on your C.W. book. = 16 o = 53 o = 67 o x = 7.07 x = 22.7 x = 3.76

29 Part 4 Do Ex.11A Q (P.180) on your C.W. book. sin 7 25 = 16 o sin 8 10 = 53 o sin = 67 o

30 Part 4 Do Ex.11A Q (P.180) on your C.W. book. x 10 sin 45 o x = 10 sin 45 o x = x x o sin o sin 25 o 5 sin 36.5 x = 22.7 x = 3.76 x x 5 o sin 36.5

31 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.