Chapter 12 Application of Trigonometry Additional Example 12.1Additional Example 12.1 Additional Example 12.2Additional Example 12.2 Additional Example.

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Chapter 12 Application of Trigonometry Additional Example 12.1Additional Example 12.1 Additional Example 12.2Additional Example 12.2 Additional Example 12.3Additional Example 12.3 Additional Example 12.4Additional Example 12.4 Additional Example 12.5Additional Example 12.5 Additional Example 12.6Additional Example 12.6 Additional Example 12.7Additional Example 12.7 Additional Example 12.8Additional Example 12.8 Additional Example 12.9Additional Example 12.9 Additional Example 12.10Additional Example Example 1Example 1 Example 2Example 2 Example 3Example 3 Example 4Example 4 Example 5Example 5 Example 6Example 6 Example 7Example 7 Example 8Example 8 Example 9Example 9 Example 10Example 10 New Trend Mathematics - S4B Quit

Chapter 12 Application of Trigonometry Additional Example 12.11Additional Example Additional Example 12.12Additional Example Additional Example 12.13Additional Example Additional Example 12.14Additional Example Additional Example 12.15Additional Example Additional Example 12.16Additional Example Additional Example 12.17Additional Example Additional Example 12.18Additional Example Additional Example 12.19Additional Example Additional Example 12.20Additional Example Example 11Example 11 Example 12Example 12 Example 13Example 13 Example 14Example 14 Example 15Example 15 Example 16Example 16 Example 17Example 17 Example 18Example 18 Example 19Example 19 Example 20Example 20 New Trend Mathematics - S4B Quit

Chapter 12 Application of Trigonometry Additional Example 12.21Additional Example 12.21Example 21Example 21 New Trend Mathematics - S4B Quit

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.1 (Correct your answers to 3 significant figures if necessary.) Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.1 According to the Pythagoras’ theorem, Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.2 In  ACD, In  ABD, Solution: In the figure, ADC is a right-angled triangle. B is a point on AC such that AB  6 cm. Find the length of BC. (Correct your answer to 3 significant figures.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.2 Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.3 In the figure,Find the area of  ABC. (Correct your answer to 3 significant figures.) Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.4 Solution: In the figure,Find the area of  ABC.

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.5 Solution: In the figure, a  12 cm, c  7 cm and B  110 . (a)Find the area of  ABC. (b)Using the result of (a), find the height of  ABC from A to BC. (c)Find b and C. (Correct your answers to 3 significant figures.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.5 Solution: (b)Let h be the height of  ABC from A to BC.

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.5 In right-angled triangle ACD, by the Pythagoras’ theorem, Solution: (c)In right-angled triangle ABD,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.6 Solution: By the sine formula, In  ABC, A  20 , C  125  and b  10 cm. Solve  ABC. (Correct your answers to 3 significant figures if necessary.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.7 By the sine formula, Solution: Solve the acute-angled triangle ABC with C  45 , b  18, c  16. (Correct your answers to 3 significant figures.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.7 Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.8 Solve  ABC if A  120 , a  16 cm and c  23 cm. (Correct your answers to 1 decimal place.) Solution: By the sine formula, There is no solution for C.

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.9 Solve  ABC if A  50 , a  9 cm and b  11 cm. (Correct your answers to 1 decimal place.) Solution:  69.4   50  < 180  and   50  < 180   There are two possible cases. By the sine formula,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example 12.9 Case 2: Case 1: Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solve  ABC if B  75 , b  32 cm and c  27 cm. (Correct your answers to 1 decimal place.) By the sine formula, Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example In  ABC, Find a. (Correct your answer to 3 significant figures.) Solution: By the cosine formula,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example If the three sides of a triangle are 13, 16 and 18, find the largest angle of the triangle. (Correct your answer to 3 significant figures.) Solution: Let a  13, b  16 and c  18. C must be the largest angle since c is the longest side. By the cosine formula,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solve  ABC if a  8, b  10 and c  13. (Correct your answers to the nearest degree.) Solution: By the cosine formula,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Find the area of each of the triangles with the sides given as follows. (Leave your answers in surd form if necessary.) (a)a  4, b  3, c  5 (b)a  3, b  6, c  7 Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example The perimeter of  ABC is 46 cm. If a : c  1 : 2 and b : c  4 : 5, (a)find a, b and c, (b)find the area of  ABC. (Correct your answer to 3 significant figures.) Solution: (a)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example (b) Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solution: In  BCD, let s be half of its perimeter. In  ABD, by the cosine formula, In the figure,  BAD  20 , AB  10 cm, AD  13 cm, CD  4 cm and BC  BD. Find the area of quadrilateral ABCD. (Correct your answer to 1 decimal place.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example A, B and C are three points on the same horizontal plane where A and B are on border L with AB  100 m. The bearings of C from A and B are S65  E and S30  E respectively. (a)Find  ACB. (b)Find the distance between B and C. (c)Find the distance of C from border L. (Correct your answers to 3 significant figures if necessary.) Solution: (a)In  ABC,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example (b)By the sine formula, Solution: (c)In  CBD,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example In the figure, PQ and TR are two vertical poles on the horizontal ground AQR. The angles of elevation of T and P from A are 35  and 20  respectively. If AR  50 m and QR  10 m, find (a)the length of PT. (b)the angle of elevation of T from P. (Correct your answers to 3 significant figures.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example (a)(a) Solution: In  APQ, In  ATR,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example In  APT, by the cosine formula, Solution: (b) In  TPS,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example A man walks 100 m in the direction 160  from A to B. Then he walks 45 m in the direction 120  from B to C. (a)Find  ABC. (b)Find the distance between A and C. (c)Find the true bearing of C from A. (Correct your answers to 3 significant figures if necessary.) Solution: (a)In right-angled triangle AQB,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit (b)By the cosine formula, Solution: Additional Example (c) By the sine formula,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solution: Since ABCDEF is a regular hexagon, we have following figure. The figure shows a right pyramid VABCDEF with regular hexagonal base ABCDEF. If VA  10 cm, AB  4 cm and P is a point on VB such that AP  VB, (a)find the angle between line VA and plane ABCDEF. (b)find the lengths of AC and AP. (c)find the angle between planes VAB and VBC. (Correct your answers to 3 significant figures.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit (a)Let  be the angle between line VA and plane ABCDEF. AO = 4 cm In  VOA, Solution: Additional Example (b) By the sine formula,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example In  VAB, by the cosine formula,  VABCDEF is a right pyramid.  VB  VA  10 cm Solution: In  APB,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example (c) In  ACP, by the cosine formula, Solution:

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example In the figure, AEF is a horizontal plane and ABC is an inclined plane. The inclinations of AB and BC are both 15 . If AB  100 m, BC  120 m and the angle between AB and AC is 60 , (a)find the height of C above horizontal plane AEF. (b)find the inclination of AC. (Correct your answers to 1 decimal place.)

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example Solution: (a)In  BCD, In  ABE,

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit (b)In  ABC, by the sine formula, Solution: By the sine formula, Additional Example 12.21

Chapter 12 Application of Trigonometry 2004 Chung Tai Educational Press © Quit Additional Example In  ACF, Solution:  The inclination of AC is 25.3 .