Oblique Triangle
To find the width of a river, a surveyor set up his transit at C on one bank and sighted across to a
point B on the opposite bank; then turning through an angle of 90', he laid off a distance CA =
225 m. Finally, setting the transit at A, he measured LCAB as 48'20'. Find the width of the river.
From a point A on level ground, the angles of elevation of the top D and bottom B of a flagpole
situated on the top of a hill are measured as 47'54' and 39'45'. Find the height of the hill if the height
of the flagpole is 115.5 ft.
From the top of a lighthouse, 175 ft above the water, the angle of depression of a boat due south is
18'50'. Calculate the speed of the boat if, after it moves due west for 2 min, the angle of depression is 14'20'.
A parked car is spotted from a hotel window which is 100 m above the car. If the angle of depression from the
window to the car is 15.4', how far is the car from the window?
A tower 125 ft high is on a cliff on the bank of a river. From the top of the tower the angle of
depression of a point on the opposite shore is 28'40' and from the base of the tower the angle of
depression of the same point is 18'20'. Find the width of the river and the height of the cliff.
Solve the triangle ABC, given c = 628, b = 48, and C = 55'10'.
From A, a pilot flies 125 km in the direction N38'20'W and turns back. Through an error, the pilot
then flies 125 km in the direction S51'40'E. How far and in what direction must the pilot now fly to
reach the intended destination A?
Solve the triangle ABC, given a = 25.2, b = 37.8, and c = 43.4.
A lighthouse is 10 km northwest of a dock. A ship leaves the dock at 9 A.M. and steams west at 12 k m h . At
what time will it be 8 km from the lighthouse?
A person, standing on the bank of a river, observes that the angle subtended by a tree on the opposite
bank is 60°; when he retreats 20m from the bank, he finds the angle to be 30°. Find the height of the tree and the
breadth of the river.
The angle of elevation of an airplane from a point on the ground is 45°. After a flight of 15 sec, the elevation
changes to 30°. If the airplane is flying at a height of 3000 meters, find the speed of the airplane.
If the length of the shadow of a tower is 3 times that of its height, then the angle of elevation of the sun is
Oblique Bearings
Three ships are situated as follows: A is 225 mi due north of C,and B is 375 mi due east of C. What
is the bearing (a) of B from A and (b) of A from B?
Three ships are situated as follows: A is 225 miles west of C while B, due south of C, bears S25'10'E
from A. (a)How far is B from A? (b)How far is B from C? (c)What is the bearing of A from B?
From a boat sailing due north at 16.5 km/h, a wrecked ship K and an observation tower Tare observed
in a line due east. One hour later the wrecked ship and the tower have bearings S34'40'E and S65'10'E.
Find the distance between the wrecked ship and the tower.
A ship is sailing due east when a light is observed bearing N62'10'E. After the ship has traveled
2250 m, the light bears N48'25'E. If the course is continued, how close will the ship approach the light?
A river flows due south at 125 ft/min. A motorboat, moving at 475 ft/min in still water, is headed
due east across the river. (a) Find the direction in which the boat moves and its speed. (6) In what
direction must the boat be headed in order that it move due east and what is its speed in that direction?
Spherical Trigo
If Greenwich Mean Time (GMT) is 6 A.M., what is the time at a place located 30° East longitude?
If the longitude of Tokyo is 139°E and that of Manila is 121°E, what is the time difference between Tokyo and
Manila?
One degree on the equator of the earth is equivalent to __ in time.
A spherical triangle ABC has an angle C =90° and sides a = 50° and c = 80° Find the value of "b" in degrees.
Solve the remaining side of the spherical triangle whose given parts are A = B =80° and a =b =89°
Solve for side b of a right spherical triangle ABC whose parts are a ::: 46°, c "' 75° and C =90°
Given a right spherical triangle whose parts are a = 82°, b = 62° and C = 90°. What is the value of the side opposite
the right angle?
Determine the value of the angle B of an isosceles spherical triangle ABC whose given parts are b = c =54°28' and a
= 92°30'.
Solve the angle A in the spherical triangle ABC, given a = 106°25', c = 42°16' and B = 114°53'. Solve for angle C of
the oblique spherical triangle ABC given, a = 80°, c = 115° and A = 72°
Determine the spherical excess of the spherical triangle ABC given a = 56°, b 65" and c =78°
What is the spherical excess of <> spherical triangle whose angles are all right angles?
The area of spherical triangle ABC whose parts are A= 9::1°40'. B = 64°12', C = i 16°51' and the radius of the spl1ere
is 100 m is:
A spherical triangle has an area of 327.25 sq. km. What is the radius of the sphere if its spherical excess is 30°? A
ship on a certain day is at latitude 20°N and longitude i 40°E. After sailing for 150 hours at a uniform speed along a
great circle route, it reaches a point at latitude 10°S and longitude 170°E. If the radius of the earth is 3959 miles.
find the speed in miles per hour.
A degree (") is defined as the measure of the central angle subtended by an arc of a circle equal to U360
A minute (') is 1/60 of a degree; a second (") is 1/60 of a minute, or 113600 of a degree.
A radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to the
radius of the circle.
QUADRANT SIGNS OF THE FUNCTIONS
ANGLES OF DEPRESSION AND ELEVATION An angle of depression is the angle from the horizontal down
to the line of sight from the observer to an object below. The angle of elevation is the angle from the
horizontal up to the line of sight from the observer to an object above.
In Fig. 3-5, the angle of depression from point A to point B is a and the angle of elevation from point B to
point A is p. Since both angles are measured from horizontal lines, which are parallel, the line of sight AB
is a transversal, and since alternate interior angles for parallel lines are equal, a = p. (See App. 1,
Geometry.)
BEARING The bearing of a point B from a point A, in a horizontal plane, is usually defined as the angle
(always acute) made by the ray drawn from A through B with the north-south line through A. The
bearing is then read from the north or south line toward the east or west. The angle used in expressing a
bearing is usually stated in degrees and minutes. For example, see Fig. 5- 1.
In aeronautics the bearing of B from A is more often given as the angle made by the ray AB with the
north line through A, measured clockwise from the north (i.e., from the north around through the east).
For example, see Fig. 5-2.