Plane trigonometry with practical applications, by Leonard E. Dickson.

128

TRIGONOMETRY

[Art. 9i

EXERCISES ON THE LAW OF COSINES
1. Using the law of cosines, write the two further equations of type
(9), and from them deduce two of type (13).
Hence solve the triangles in which
2. a = 2, b = 3,c = 4. 3. a = 7,b = 8, c = 9.
4. a = 2, b = 3, C= 30~. 5. b = 1, c = 10, A = 35024'.
6. The two diagonals of a parallelogram are 6 and 8, and their included angle is 60~. Find the sides.
7. Find the inclination of the face of an embankment if a ladder 26
feet long rests 10 feet from the foot of the embankment and reaches 23
feet up its face.
8. From the crossing, at an angle of 54~, of two railroads the distances
to two bridges over the same river are 4 and 6 miles. How far apart are
the bridges?
9. Two towns A and B are 15 and 20 miles distant from C, while angle
ACB is 27~30'. Two men start at the same instant from C and walk at
the same rate. One walks to A and then toward B. The other walks to
B and then toward A. How far from A do they meet?
10. Divide each member of (10) by b sin A and replace a lb by its value
given by the law of sines. Hence show that
C
cot B = - cot A.
b sin A
By this formula compute B, given b = 8, c = 12, A = 30~.
11.* The plane ABC of a side of a hill is inclined 30~ to the horizontal
plane ABD and intersects the latter in an east and west line AB. A tree
standing at A had its upper part broken at a point 12 feet above the
ground by a WSW wind (i.e., blowing from S 67 ~ W), and the top of the
broken tree now rests on the ground 40 feet from A. Find the height of
the tree.
91. Area (A) of a triangle in terms of its sides.1 By Art. 86,
A = ~bc sin A, A2 = b2c2 sin2 A = b2c2 (1-cos 2 A).
We shall evaluate the two factors 1 + cos A, 1 - cos A. By (13),
- c (b+c)2-a2 a2 - (b-c)2
1 + cos A - a, 1- cos A=
2be 2bc
1The instructor may assume this result from geometry.