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G.S.C.- EG, Math 1. Differential Calculus & Trigonometry , Trigonometry , CHAPTER One- Exercises And Examination

G.S.C.- EG, Math 1. Differential Calculus & Trigonometry , Trigonometry , CHAPTER One- Exercises And Examinations (Free Educational Material) Prepared by: Dr. Ashraf Aboshosha http://www.icgst.com/A_Aboshosha.html editor@icgst.com Tel.: 0020-122-1804952 Fax.: 0020-2-24115475.

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G.S.C.- EG, Math 1. Differential Calculus & Trigonometry , Trigonometry , CHAPTER One- Exercises And Examination

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  1. G.S.C.- EG, Math 1. Differential Calculus & Trigonometry, Trigonometry, CHAPTER One- Exercises And Examinations (Free Educational Material) Prepared by: Dr. AshrafAboshosha http://www.icgst.com/A_Aboshosha.html editor@icgst.comTel.: 0020-122-1804952Fax.: 0020-2-24115475

  2. Exercises

  3. Exercise 1-1 • In the triangle ABC: • In the triangle ABC: • In the parallelogram ABCD: • LMN is a triangle in which:

  4. Exercise 1-1 • In any triangle ABC, prove that its area is equal to 2r2*sinA*sinB*sinC. • ABC is a triangle in which: find c and the length of the radius of the circumcircle of the triangle ABC. • ABC is an obtuse angled triangle at A in which: find a , c , and the area of the triangle to the nearest integer.

  5. Exercise 1-1 • Find the perimeter of the triangle ABC in which: • ABC is a triangle in which: if its perimeter equals 12cm, find AB to the nearest integer.

  6. Exercise 1-2 • ABC is a triangle in which: a=283cm, b=317cm and c=428cm, find the measure of its angles. • ABC is a triangle in which: • ABC is a triangle in which: • ABC is a triangle in which:

  7. Exercise 1-2 • In the triangle ABC: • In the triangle ABC: • ABCD is a parallelogram in which:

  8. Exercise 1-2 • In the parallelogram ABCD, prove that: • ABC is a triangle. D is the mid-point of BC. Prove that: • ABCD is a parallelogram in which the diagonals have lengths 5cm and 8cm. And the measure of the angle between them is 77o18\. Find the length of AB and AC.

  9. Exercise 1-3 • Solve the triangle ABC in the following cases:

  10. Exercise 1-4 • The elevation angle of the top of a flag was measured from a point at a distance 50m. From the base of the flag, and found to be 30o. Find the height of the flag. • A tree was broken such that its top touched the ground its upper part made an angle of measure 60o with the ground. If the point of the contact of the top of the tree with the ground was at a distance 10m from its bottom, find the length of the tree. • A tower of height 50m is constructed on a hill. From a point on the ground level the measure of the elevation angles of the top and the base of the tower were 75o and 45o respectively. Find the height of the hill.

  11. Exercise 1-4 • At a certain time the depression angles of two ships A and B were measured from a flying airplane at a height 600 meters above the sea surface were found to be 45o,35o respectively. If the projection of the airplane belongs to AB, find the distance between the two ships A and B to the nearest meter. • From a widow of a building 6m high, the elevation angle of the top of a tree was measured and found to be 15o, and the measure of depression angle of the bottom of that tree was 30o. Find the height of the tree and its distance from the building.

  12. Exercise 1-4 • A man observed the top of a tower at an angle of elevation of 25o. When he walked horizontally towards the base of the tower 40m, the measure of the angle of elevation of the top becomes 42o. Find the height of the tower to the nearest metre. • From a boat moving away from a rock, the elevation angle of the top of the rock was measured at a certain moment and found to be 60o . After two minutes, the measure of the elevation angle has become 45o. Find the velocity of the boat if the height of the rock is 500m above the sea level.

  13. Exercise 1-4 • The horizontal distance between two towers A and B is 60m. The height of B is 150m . If the measure of depression angle of the top of A (observed from the top of B) is 30o, find the height of A.

  14. Exercise 1-5 • Solve the triangle ABC in the following cases: • If the side length of a triangle are x2+x+1, 2x+1 and x2-1, prove that its largest angle is a measure of 120o. • Prove that the angle of measure 76o39\ is the angle of the greatest measure in the triangle whose side lengths are 242cm, 188cm and 270cm.

  15. Exercise 1-5 • A man standing at a point B observed an object C in the east of B and at a distance 60m from B. After the man walked from B to A in the direction 60o north of the east; he found that the point C is in the distance 15o south of the east from A. Find the distance CA. • From a point A on the edge of a river, a man observed a point B on the opposite parallel edge of the river and he found it at 20o north of the east. When the man walked 300m along the edge to point C, he found B to be at 46o north of the east from C. Find the width of the river, given that the points A, B and C are in the same horizontal level.

  16. Examinations

  17. Exam 1995 • First session • ABC is a triangle in which: find the perimeter of the triangle and the length of the radius of the circle which passes through the vertices of the triangle. • From a point on the ground level, a man found that the measure of the elevation angle of the top of a tower is 20o35\ when he walked 50m on a horizontal road towards the base of the tower, he found that the elevation angle of the top of the tower is 42o.Find the height of the tower to the nearest meter • Find the measure of the smallest angle in the triangle XYZ given that :

  18. Exam 1995 • Second session • If the lengths of the sides of a triangle are 7cm, 3cm and 5cm. Find the measure of its greatest angle. • Find the perimeter of the triangle ABC in which c=8.7cm, • From a point at a distance 100m. From the base of a tower under construction the measure of elevation angle was 30o how many meters should the top of the tower raise for the angle of elevation of the top from the same point to be 45o.

  19. Exam 1996 • First session • In the triangle ABC if a=12cm, b=13cm and c=10cm. Find the measure of the angle A to the nearest minute, then determine the length of the radius of the circle which passes through it vertices. • From the top of a house 15m high the measure of elevation angle of the top of the tower is 67o and the measure of depression of its base is 35o. Find the height of the tower to the nearest meter. Where the base of the tower and the base of the house are in the same horizontal plane. • ABCD is a parallelogram, M is the intersection point of its diagonals if , find the length of AD to the nearest cm.

  20. Exam 1996 • Second session • From the top of a tower 65m height, the depression angle of two points A and B on the ground were 32o and 21o12\ respectively. If the point D represents the base of the tower and find the length of AB to the nearest meter. • ABC is a triangle in which , and BC=20 cm. D is the mid-point of BC find the length of AB and AD approximated to two decimal places.

  21. Exam 1997 • First session • ABC is a triangle in which: find the length of BC then evaluate the circumference of the circumcircle of ∆ABC • XYZ is a triangle in which: find z to the nearest millimetre . • From a point on the ground the elevation angle of the tower is observed to be 25o in the measure. if the observer walked 57m in the straight line towards the base of the tower horizontally then he found the measure of the elevation angle to be 52o30\. Find the height of the tower to the nearest metre.

  22. Exam 1997 • Second session • A lighthouse of height 60m is constructed on a hill near a sea-shore. The two elevation angle of the top and the bottom of the lighthouse are measured from a boat on the sea level to be found 70o and 45o respectively. Find the height of the hill above the sea level to the nearest metre. • ABCD is a parallelogram in which: calculate the length of the diagonal AC to the nearest cm. • ABC is a triangle in which prove that: ∆ABC is isoscales.

  23. Exam 1998 • First session • ABC is a triangle in which: b=15cm, find c and the length of the radius of the circumcircle of the triangle ABC. • From the top of a rock of height 80 metres. A man found that the measure of two depression angles of the top and the base of a tower were 24o, 35o respectively. Find the height of the tower to the nearest metre knowing that the base of the rock and the base of the tower were in the same horizontal plane .

  24. Exam 1998 • Second session • ABC is a triangle in which: find c. • From a point on the ground level a man observed the elevation angle of the top of the tower. He found its measure equals 35o. if the observer walked 30m in the straight line towards the base of the tower horizontally then he found the measure of the elevation angle to be 46o. Find the height of the tower to the nearest metre.

  25. Exam 1999 • First session • ABC is a triangle in which: , find b, and the length of the radius of the circumcircle of ∆ ABC approximated to two decimals. • XYZ is a triangle in which: x=15cm, y=12cm, and find z to the nearest cm. • From a point on the ground level, the elevation angle of the top of a tower was measured and found to be 27o when the observer walked horizontally towards the base of the tower 55m, the measure of the elevation angle of the tower’s top became 56o find the height of the tower to the nearest m.

  26. Exam 1999 • Second session • Find the measure of the smallest angle in the triangle ABC given that a=7cm, b=5cm, and c=10cm. • XYZ is a triangle in which: find x to the nearest cm. and the area of the triangle XYZ to the nearest cm2 • From the top of a hill, a man found that the depression angles of the top and the base of a tower were 22o and 30o respectively. If the height of the tower is 50m, calculate the height of the hill to the nearest metre.

  27. Exam 2000 • First session • Find the measure of the greatest angle in the triangle ABC in which a=8cm, b=7cm, and c=13cm. • Solve the triangle ABC in which: approximating the length to the nearest cm. • A tower of height 100 meters stands on a hill. From a point on the ground level in the same horizontal plane containing the base of the hill, the measure of the elevation angles of the top and the base of the tower are 76o and 46o respectively. Find the height of the hill to the nearest meter.

  28. Exam 2000 • Second session • ABC is a triangle in which: a=13cm, b=14cm, and c=15. find , then calculate the area of the triangle ABC to the nearest cm2. • Solve the triangle ABC in which b=11cm, approximating the length to the nearest cm. • From the top of a tower of height 74m the depression angles of the top and the base of a building are measured and found to be 25o and 36o respectively. Find the height of the building to the nearest meter knowing that the base of the tower and the building lie in the same horizontal plane.

  29. Exam 2001 • First session • ABC is a triangle in which a=3cm, b=6cm , find c to the nearest cm. • LMN is a triangle in which L=4.2cm, . find n and the length of the diameter of the circumcircle of ∆LMN to the nearest one decimal. • From a point on the ground the elevation angle of the top of the tower was measured and found to be 34o when the observer walked on a horizontal straight line towards the base of the tower a distance of 62m the measure of the elevation angle of the top of the tower became 51o. Find the height of the tower to the nearest meter.

  30. Exam 2001 • Second session • Find the measure of the smallest angle in the triangle XYZ in which x=28cm, y=17cm and z=20cm. • Find the perimeter of triangle ABC in which: c=9cm, to the nearest one integer. • From the top of a mountain an observer found that the measure of the depression angles of the top and the base of a tower are 15o and 26o respectively. If the height of the tower is 20m. Calculate the height of the mountain to the nearest meter, given that the two bases of the mountain and the tower are in the same horizontal plane.

  31. Exam 2002 • First session • ABC is a triangle in which a=6cm, b=8cm, and find the perimeter of this triangle to the nearest cm. • ABC is an isosceles triangle in which and the length of the radius of the circumcircle of ∆ ABC is equal to 20cm. Find b and then deduce the area of ∆ABC to the nearest cm2 • From a point A on a side of a river, a man observed a house at a point B on the other side of the river at 20o north east, when the man walked a distance of 300m, parallel to the side of the river towards the east he arrived at a point C from where he found that the point B became at a direction of 46o north east . Find the width of the river to the nearest meter , knowing that A, B, and c are in the same horizontal plane.

  32. Exam 2002 • Second session • ABC is a triangle in which and a=3.5cm, calculate the perimeter of ∆ABC to the nearest cm. • ABC is a triangle in which a=20cm, b=12cm, . Calculate c to the nearest cm. • From the top of a house of height 24m the measure of the elevation angle of the top of a tower equals 15o and the measure of depression angle of the tower equals 30o find the height of the tower and its distance from the house to the nearest meter if known that the two bases of the house and the tower are in the same horizontal plane.

  33. Exam 2003 • First session • ABC is a triangle in which a=13cm, b=14cm, and c=15cm. Find , then find the area of the triangle ABC. • If the perimeter of the triangle ABC equals 12cm, find the length of AB to the nearest cm. • From the top of a tower, it is found that the depression angles of the top and the base of a minaret are 32o and 57o respectively. If the height of the minaret is 37m, find the distance between the base of the tower and of the minaret to the nearest m, given that the base of the tower and that of the minaret lie in the same horizontal plane.

  34. Exam 2003 • Second session • ABC is a triangle in which a=13cm, b=14cm, and c=15cm. Calculate the measure of the smallest angle. • From the top of a tower with height 62m above the surface of the ground , the depression angles of two points X,Y on the ground in the same side of the tower were measured to be found 18o and 43o respectively if passes through the base of the tower, find the length of XY. • ABC is a triangle in which: and the length of the radius of the circumcircle of ∆ABC equals 20cm. Calculate the area of ∆ABC to the nearest cm2.

  35. Exam 2004 • First session • Solve the triangle XYZ in which: , XZ=16cm and YZ=13cm. • ABC is a triangle in which a=20cm, . Find the area of this triangle. • A boat moves in a straight line towards a rock with velocity 300m/min. the elevation angle of the top of the rock is measured from the boat at a certain moment and found to be 35o, after two minutes and from the same boat the measure of the elevation angle has become 60o find the height of the rock to the nearest meter.

  36. Exam 2004 • Second session • ABC is a triangle in which b=5cm, c=6cm and . Find the values of: • From the top of a tower with height 75m. The depression angles of the top and the base of a house were measured to be 24o,35o respectively. Find the height of the house to the nearest meter if known that the base of the tower and the house are in the same horizontal plane. • ABC is a triangle in which , and b=3cm find the area of the circumcircle of ∆ABC.

  37. Exam 2005 • First session • ABC is a triangle in which a=6cm, find c to the nearest cm. • Solve the triangle XYZ in which x=20cm, y=12cm, and • From the top of a mountain it was found that the measure of the depression angles of the top and the base of a tower are 35oand 46o respectively . If the height of the mountain equals 125m find the height of the tower to the nearest meter given that the base of the tower and that of the mountain lie in the same horizontal plane.

  38. Exam 2005 • Second session • Solve the triangle ABC in which a=15cm, b=13cm, and c=14cm. • ABC is a triangle in which and b=3cm. Find c and the radius of the circumcircle of ∆ ABC. • A ship started sailing from a fixed point in the direction 12o south of the east with velocity 11Km/h at the same time, another ship started sailing from the same point in direction 68o north of the east with velocity 6.5 Km/h find the distance between the two ships after two hours from the moment of starting sailing.

  39. Exam 2006 • First session • Solve the triangle ABC in which a=13cm, and the length of the radius of the circumcircle of ∆ABC is 8cm. • .In the figure: two balloons A,H at a height of , 50m observes an object C on the ground located in the vertical plane containing the two balloons. If the measure of depression angles of the object are 45o,30o respectively find the distance between the two balloons to the nearest meter. A H 141.42m 50m D C B

  40. Exam 2006 • Second session • ABC is a triangle in which , b=12cm. Find c to the nearest cm. • Find the measure of the greatest angle of a triangle ABC in which a=8cm, b=7cm, and c=9cm. • From a point on the ground a man observed the top of a tower at an angle of elevation of 25o when he walked horizontally on a straight line towards the base of the tower 50m, the measure of the angle of elevation of the top of the tower became 40o. Find the height of the tower to the nearest meter.

  41. Exam 2007 • First session • ABC is a triangle in which , AC=10cm. Find the length of AB, and the length of the radius of the circumcircle of the triangle ABC to the nearest cm. • LMN is a triangle in which L=10cm, m=9cm, and n=8cm. Calculate , then find the area of the triangle LMN. • At the same time from a port, two ships started sailing, the first in the direction 50o north of the east with velocity 8Km/h, and the second in the direction 25o south of the east with velocity 15Km/h. find the distance between the two ships after two hours from the moment they started to sail together to the nearest Km.

  42. Exam 2007 • Second session • ABC is a triangle , in which a=12cm, b=13cm, and c=14cm. Find , then calculate the area of the triangle ABC. • XYZ is a isosceles triangle, in which ,YZ=10cm. Find the perimeter of the triangle and the length of the radius of the circumcircle of the triangle. • From a balcony of a building of 8m high from the surface of the ground, the elevation angle of the top of a tree was measured and found to be 15o, and the measure of the depression angle of the bottom of that tree was 25o. Find the height of the tree and its distance from the building .

  43. Exam 2008 • First session • ABC is a triangle in which a=8cm, b=7cm, and . Find to the nearest cm the values of: c and the length of the radius of the circumcircle of the triangle. • From a point on the horizontal land, a man found that the measure of the elevation angle of a balloon moving vertically with velocity 20m/min. is 35o. After 3 minutes and from the same point the man found that the measure of the elevation angle of the balloon became 15o. Find the distance between the man and the projection of the balloon on the land to the nearest meter.

  44. Exam 2008 • Second session • ABC is a triangle in which a=8cm, . Find the values of: the length of the radius of the circumcircle of the triangle ABC, and the area of the triangle to the nearest cm2. • A tower of height 70m stands on a rock. From a point on the ground level, the measure of elevation angles of the top and the base of the tower are 65o and 33o respectively. Find the height of the rock to the nearest meter. • ABC is a triangle in which 10a=12b=15c. Find the measure of the smallest angle.

  45. Exam 2009 • First session • Solve the triangle XYZ in which x=10cm, and • ABC is a triangle in which a=4cm, and the area of this triangle= . Find b. • From a point on the ground level, the elevation angle of the top of a tower whose base is on the same ground level was measured and found to be 22o. When the observer walked on a horizontal straight line towards the base of the tower a distance of 50 meter, the measure of the elevation angle of the top of the tower became 36o. Find the height of the tower to the nearest meter.

  46. Exam 2009 • Second session • Find the measure of the smallest angle of the triangle ABC given that a=7cm, b=5cm, and c=10cm. • ABC is a triangle in which , and AC=10cm. Find AB then the length of the radius of the circumcircle of the triangle ABC. • From the top of a tower 75m height, the angles of depression of the top and the base of a house were 35o and 48o respectively. Find the height of the house to the nearest meter, given that the base of the tower and the base of the house are in the same horizontal plane,

  47. Exam 2010 • First session • ABC is a triangle in which a=8cm, b=5cm, and . Calculate the surface area of the circumcircle of the triangle ABC to the nearest cm2. • A horizontal road joins the bases of a tower and a house. From a point on the road which lies between the two bases. A man observed the two elevation angle of the top of the tower and the top of the house, he found the measure of these two angles are 75o and 43o respectively. If it known that the tower, the house and the road are in the same vertical plane find the distance between the top of the house and the top of the tower to the nearest meter. given that the height of the tower is 100m and the height of the house is 35m

  48. Exam 2010 • Second session • In the triangle ABC in which a=6cm, b=10cm and c=7cm. Find the measure of the smallest angle in the triangle. • If the perimeter of the triangle ABC equals 12cm, then find the length of AB to the nearest cm. • From the top of a tower it was found that the depression angles of the top and the base of a house are 32o and 57o respectively. If the height of the house is 27m. Find the distance between the base of the tower and the base of the house to the nearest meter. Given that the two bases lie in the same horizontal plane.

  49. Exam 2011 • First session • From a point on the ground, a man found that the measure of the elevation angle of a tower top is 25o , then when he covered a distance of 45m towards the base of the tower , he found that the measure of the elevation angle of the tower top became 65o. Find the height of the tower to the nearest meter. • ABCD is a parallelogram in which its perimeter 20cm. If , BD=8cm, calculate the length of each AB , AD.

  50. Student’s Evaluation Guide Exersices

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