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129. 1. From a point in the same horizontal plane with the base of a tower, the angle of elevation of its top is 52° 39', and from a point 100 feet further away it is 35° 16'. Required the height of the tower, and its distance from each of the points of observation.

2. In a field ABCD, the sides AB, BC, CD, and DA are 155, 236, 252, and 105 rods, respectively, and the length of a line from A to C is 311 rods. Find the area of the field.

3. From the top of a bluff, the angles of depression of two posts in the plain below, in line with the observer and 1000 feet apart, are found to be 27° 40' and 9° 33', respectively. What is the height of the bluff above the plain?

4. Two yachts start at the same time from the same point, and sail, one due north at the rate of 10.44 miles an

hour, and the other due northeast at the rate of 7.71 miles an hour. How far apart are they at the end of 40 minutes?

5. A ship is sailing due southwest at the rate of 8 miles an hour. At 10.30 A.M., a lighthouse is observed to bear 30° west of north, and at 12.15 P.M., it is observed to bear 15° east of north. Find the distance of the lighthouse from each position of the ship.

6. Wishing to find the distance of an inaccessible object A from a position B, I measure a line BC, 208.3 feet in length. The angles ABC and ACB are measured, and found to be 126° 35' and 31° 48', respectively. Required the distance AB.

7. A flagpole 40 feet in height stands on the top of a tower. From a position near the base of the tower, the angles of elevation of the top and bottom of the pole are 38° 53′ and 20° 18', respectively. Required the distance and height of the tower.

8. A surveyor observes that his position A is exactly in line with two inaccessible objects B and C. He measures a line AD, 500 feet in length, making the angle BAD = 60°, and at D observes the angles ADB and BDC to be 40° and 60°, respectively. Required the distance BC.

9. To find the distance between two buoys A and B, I measure a base line CD on the shore, 150 feet in length. At the point C the angles ACD and BCD are measured and found to be 95° and 70°, respectively; and at D the angles BDC and ADC are found to be 83° and 30°. What is the distance between the buoys?

10. The sides of a field ABCD are AB=37, BC=63, and DA = 20, and the diagonals AC and BD are 75 and 42, respectively. Required the area of the field.





130. If a triedral angle is formed with its vertex at the centre of a sphere, it intercepts on the surface a spherical triangle.

131. The triangle is bounded by three arcs of great circles called its sides, which measure the face angles of the triedral angle.

The angles of the spherical triangle are the diedral angles of the triedral angle; and by Geometry, each is measured by the angle between two straight lines drawn, one in each face, and perpendicular to the edge at the same point.

132. The sides of a spherical triangle, being arcs, are usually expressed in degrees.

If the length of a side in terms of some linear unit is desired, it may be obtained by finding the ratio of its are to 360°, and multiplying the result by the length of the circumference of a great circle.

133. Spherical Trigonometry treats of the trigonometrical relations between the elements of a spherical triangle; or what is the same thing, between the face and diedral angles of the triedral angle which intercepts it.

134. The face and diedral angles are not altered in magnitude by varying the radius of the sphere, and hence the relations between the sides and angles of a spherical triangle are independent of the length of the radius.

135. We shall limit ourselves in this work to such triangles as are considered in Geometry, where each angle is less than two right angles, and each side less than the semicircumference of a great circle; that is, where each element is less than 180°.

136. The proofs of the following properties of spherical triangles may be found in any treatise on Solid Geometry:

(a) Either side of a spherical triangle is less than the sum of the other two sides.

(b) If two sides of a spherical triangle are unequal, the angles opposite them are unequal, and the greater angle lies opposite the greater side; and conversely.

(c) The sum of the sides of a spherical triangle is less than 360°.

(d) The sum of the angles of a spherical triangle is greater than 180°, and less than 540°.

(e) If A'B'C' is the polar triangle of ABC, i.e., if A, B, and C are the poles of the arcs a', b', and c', respectively, then conversely, ABC is the polar triangle of A'B'C'.

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(f) In two polar triangles, each angle of one is measured by the supplement of the side lying opposite to it in the other.

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137. A spherical triangle is called tri-rectangular when it has three right angles; each of its sides is a quadrant, and each vertex is the pole of the opposite side.

138. I. Let C be the right angle of the 'spherical right triangle ABC, and suppose a < 90° and b < 90°.

Complete the tri-rectangular triangle A'B'C.

Also, since B' is the pole of AC, and A' of BC, construct the tri-rectangular triangles AB'D and A'BE.

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Then since A and B lie on the same side of B'D, AB or c is < 90°.

Since BC is < B'C, the angle A is < B'AD, or < 90°. Since AC is < A'C, the angle B is < A'BE, or < 90°.

II. Suppose a < 90° and b > 90°.

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