Plane trigonometry, by S.L. Loney.

466

TRIGONOMETRY.

[Exs. LXVI.]

angles ADB and CDB are equal and of an observed magnitude 0; prove
that the error in the calculated length of DB consequent on a small
error 6 in the observed magnitude of 0, is
2ab (a+b)2sin 0
(a2 + b2 - 2ab cos 20)A
approximately, where AB=a and BC=b.
7. In measuring the three sides of a triangle small errors x and y
are made in two of them, a and b; prove that the error in the angle C
will be- Y cot A - cot B, and find the errors in the other angles.
b a
8. In a triangle ABC we have given that approximately a=36 feet,
3
b=50 feet, and C=tan-' '; find what error in the given value of a will
cause an error in the calculated value of c equal to that caused by an
error of 5" in the measurement of C.
9. A triangle is solved from the parts C=15~, a =/6, and b=2;
prove that an error of 10" in the value of C would cause an error of about
13'66" in the calculated value of B.
10. Two sides b and c and the included angle A of a given triangle
are supposed to be known; if there be a small error 0 in the value of the
angle A, prove that
(1) the consequent error in the calculated value of B is
- 0 sin B cos C cosec A radians,
(2) the consequent error in the calculated value of a is c sin B. 0,
and (3) the consequent error in the calculated area of the triangle is
0 cot A times that area.
11. There are errors in the sides a, b, and c of a triangle equal to
x, y, and z respectively; prove that the consequent error in the calculated
value of the circum-radius is
cot A cot B cot C [x sec A +y sec B +z sec C].
2
12. The area of a triangle is found by measuring the lengths of the
sides and the limit of error possible, either in excess or defect, in
measuring any length is n times that length, where n is small. Prove that
in the case of the triangle whose sides are measured as 110, 81, and
59 yards, the limit to the error in the deduced area of the triangle is
about 3'1433n times that area.