16.09 Trigonometry in 3D (Extended)
All edges of the following cube are 7 \text{ cm} long:
Find the exact length of the following:
EG
AG
Find the following angles to the nearest degree:
\angle EGH
\angle EGA
\angle AHG
\angle AGH
A square prism has sides of length 3 \text{ cm}, 3 \text{ cm} and 14 \text{ cm} as shown in the diagram:
If the diagonal HF has a length of z \text{ cm}, find the value of z to two decimal places.
If the size of \angle DFH is \theta \degree, find \theta to two decimal places.
The box have a triangular divider placed inside it, as shown in the diagram:
If z = AC, find the value of z to two decimal places.
Find the area of the divider correct to two decimal places.
A triangular prism has dimensions as shown in the diagram:
Find the size of \angle AED, correct to two decimal places.
Find the exact length of CE.
Find the exact length of BE.
Find the size of \angle BEC, correct to two decimal places.
Find the exact length of CX.
Find the size of \angle BXC, correct to two decimal places.
Consider the given rectangular prism:
Find the length x.
Hence, find the length of the prism's diagonal y.
Find the angle \theta to the nearest degree.
Consider the cube as shown:
Find the size of:
\alpha
\beta
\gamma
Calculate the exact length of x.
Calculate the exact length of y.
Hence, find the angle \theta to the nearest degree.
A 25 \text{ cm }\times 11 \text{ cm }\times 8 \text{ cm} cardboard box contains an insert (the shaded area) made of foam:
Find the exact length b of the base of the foam insert.
Find the area of foam in the insert, rounded to the nearest square centimetre.
Find the value of \theta to the nearest degree.
A tree stands at the corner of a square playing field. Each side of the square is 130 \text{ m} long. At the centre of the field the tree subtends an angle of 22 \degree as shown:
Find the distance from the tree to the centre of the field. Round your answer to two decimal places.
Find the height of the tree. Round your answer to two decimal places.
Find the size of the angle subtended by the tree at the corner of the field opposite the tree. Round your answer to the nearest minute.
Find the size of the angle subtended by the tree at an adjacent corner of the field. Round your answer to the nearest minute.
A, \,C, and X are three points in a horizontal plane and B is a point vertically above X as shown in the diagram:
Find the following lengths to the nearest metre:
BC
XB
A cone has radius 7\text{ cm} and a slant height of 13\text{ cm}:
Find the vertical angle, \theta, at the top of the cone in degrees and minutes.
The following is a right pyramid on a square base with side length 12 \text{ cm}. The edge length VA is 23 \text{ cm}.
If z = AW, calculate the length z.
If \theta = \angle VAW, find the value of \theta.
The following pyramid has a square base with side length 8 \text{ cm}, and all slanted edges are 12 \text{ cm} in length:
Find the exact length of MD.
Find the size of \angle PDM, correct to two decimal places.
Find the exact height of the pyramid, MP.
Find the length of MN.
Find the exact length of PN.
Find the size of \angle PNM, correct to two decimal places.
Find the size of \angle PDC, correct to two decimal places.
This triangular prism shaped box needs a diagonal support inserted as between A and F as shown:
AB = 19, \, BD = 30 and DF = 43. find the length of AF to two decimal places.
From a point 15\text{ m} due north of a tower, the angle of elevation of the tower is 32 \degree.
Find the height of the tower h. Round your answer to two decimals places.
Find the size \theta of the angle of elevation of the tower at a point 20\text{ m} due east of the tower. Round your answer to the nearest degree.
A pole is seen by two people, Jenny and Matt:
Matt is x\text{ m} from the foot of the pole. Find x to the nearest metre.
Find the height of the pole h to the nearest metre.
Two straight paths to the top of a cliff are inclined at angles of 24 \degree and 21 \degree to the horizontal:
If path A is 115\text{ m} long, find the height h of the cliff, rounded to the nearest metre.
Find the length x of path B, correct to the nearest metre.
Let the paths meet at 46 \degree at the base of the cliff. Find their distance apart, y, at the top of the cliff, to the nearest metre.
A pyramid has a square base with side length 14 \text{ m} and a vertical height of 24 \text{ m}:
Find the length of the edge AE. Round your answer to one decimal place.
Find the size of \angle BEA, to the nearest minute.
Two buoys, A and B, are observed from a lookout at the top of a 130 \text{ m} high cliff. The bearing of buoy A is 337 \degree and its angle of depression is 3 \degree. The bearing of buoy B is 308 \degree and its angle of depression is 5 \degree:
Find the distance, to the nearest metre, from buoy A to the base of the cliff.
Find the distance from buoy B to the base of the cliff, to the nearest metre.
Find the distance between the two buoys, to the nearest metre.
In the tetrahedron shown, the angles \angle VBC, \, \angle VBA, and \angle ABC are all right-angles.
Find the following distances, to two decimal places:
VA
VC
AC
Find the size of angle \angle VCA. Round your answer to the nearest minute.
Roald is standing at point P and observes two poles, AB and CD, of different heights. P, \, B, and D are on horizontal ground:
From P, the angles of elevation to the top of the poles at A and C are 29 \degree and 18 \degree respectively. Roald is 16 \text{ m} from the base of pole AB. The height of pole CD is 7 \text{ m}.
Calculate the distance from Roald to the top of pole CD, to two decimal places.
Calculate the distance from Roald to the top of pole AB, to two decimal places.
Calculate the distance between the tops of the poles, to two decimal places.
Point A is due north of a tower, and has an angle of elevation to the top of the tower of 51 \degree. Point B is 100 \text{ m} from point A on a bearing of 120 \degree. The angle of elevation from point B to the top of the tower is 25 \degree.
Find the height of the tower to the nearest metre.
A room measures 5 \text{ m} in length and 4 \text{ m} in width. The angle of elevation from the bottom left corner to the top right corner of the room is 57 \degree.
Find the exact distance from one corner of the floor to the opposite corner of the floor.
Find the height of the room. Round your answer to two decimal places.
Find the angle of elevation from the bottom corner of the 5 \text{ m} long wall to the opposite top corner of the wall. Round your answer to two decimal places.
Find the angle of depression from the top corner of the 4 \text{ m} long wall to the opposite bottom corner of the wall. Round your answer to two decimal places.
A and B are two positions on level ground. From an advertising balloon at a vertical height of 740 \text{ m}, \, A is observed in an easterly direction and B at a bearing of 159 \degree. The angles of depression of A and B, as viewed from the balloon, are 30 \degree and 20 \degree, respectively.
Find the distance from point A to the point directly below the advertising balloon, to the nearest metre.
Find the distance from point B to the point directly below the advertising balloon, to the nearest metre.
Find the distance between points A and B, to the nearest metre.