MPM 2D1 Grade 10 Academic
SOLVING 3-D TRIGONOMETRY PROBLEMS
Conventions for reporting directions
The true bearing to a point is the angle measured in degrees in a
clockwise direction from the north line. We will refer to the true bearing
simply as the bearing.
For example, the bearing of the point P is 𝟔𝟓𝟎 , and the bearing of the
point Q is 𝟑𝟎𝟎𝟎 .
The quadrant bearing of a point is stated as the number of degrees east
or west of the north-south line. We will refer to the quadrant bearing
simply as the direction.
For example, in the diagram below,




the direction of A from O is N 𝟑𝟎𝟎 E;
the direction of B from O is N 𝟔𝟎𝟎 W;
the direction of C from O is S 𝟕𝟎𝟎 E;
the direction of D from O is S 𝟖𝟎𝟎 W;
Solve the following problems. Provide a well-labeled diagram and show
all of your work.
1. Find the value of x to the nearest centimetre.
2. Find the value of ∠BAC the nearest degree.
3. Lisa is on a 50 m high bridge and sees two boats anchored below.
From her position, boat A has a bearing of 𝟐𝟑𝟎𝟎 and boat B has a
bearing of 𝟏𝟐𝟎𝟎. Lisa estimates the angles of depression to be 𝟑𝟖𝟎 for
boat A and 𝟑𝟓𝟎 for boat B. How far apart are the boats, to the nearest
metre?
4. Marshall and Ben wish to determine the altitude, h, of a promotional
blimp. Their measurements are shown in the sketch below. What is the
altitude of the blimp, h, to the nearest of a metre?
5. To determine the height of a pole across a highway, Simone takes two
measurements. She stands at point A, directly across from the base of the
pole, B, and determines that the angle of elevation to the top of the pole
is 𝟏𝟓. 𝟑𝟎 . She then walks 30 m parallel to the highway to point C, where
she sees that the base of the pole and point A are 𝟓𝟕. 𝟓𝟎 apart. From
point A, the base of the pole and point C are 𝟗𝟎. 𝟎𝟎 apart. Calculate the
height of the pole, to the nearest metre.
6. The observation deck of the Skylon Tower in Niagara Falls is 166 m
above the Niagara River. Ashley is in the observation deck, she notices
two boats on the water. Ashley estimates that, from her position,
 the bearing of boat A is 𝟏𝟖𝟎𝟎 at an angle of depression of 𝟒𝟎𝟎 ;
 The bearing of boat B is 𝟐𝟓𝟎𝟎 at an angle of depression of 𝟑𝟒𝟎 .
Calculate the distance between the two boats, to the nearest metre.
7. While on vacation in Jamaica, Matt observes the following situation:
A coast guard helicopter hovers between an island and a damaged
sailboat. From the island, the angle of elevation to the helicopter is 𝟕𝟑𝟎 .
From the helicopter, the island and the sailboat are 𝟒𝟎𝟎 apart. A police
rescue boat bearing toward the sailboat is 800 m away from the scene of
the accident. From the police rescue boat’s position, the angle between
the island and the sailboat is 𝟑𝟓𝟎 . Matt, who is on the island, notices that
the sailboat and police rescue boat are 𝟔𝟖𝟎 apart.
Find the straight-line distance, to the nearest metre, from the helicopter
to the sailboat.
8. Cecilla would like to determine the height of her favourite tree. From
points A and B on the ground, Cecilla determines that the angles of
elevation to the top of the tree are 𝟒𝟏𝟎 and 𝟓𝟐𝟎 , respectively. The angle
formed at the base of the tree between points A and B is 𝟗𝟎𝟎 , and A and
B are 30 m apart. If Cecilla’s tree is perpendicular to the ground, what is
its height to the nearest metre?
Answers
1.
2.
3.
4. 143.4 m
5. 13 m
6. 258 m
7. 736 m
8. 22 m