Plane trigonometry, by S.L. Loney.

[Exs. XL.]

EXAMPLES.

265

Solve approximately the equations
7. sin0='01. 8. sin 0=-48.
9. cos (+ 0)=49. 10. cos 0=999.
11. Find approximately the distance at which a halfpenny, which is
an inch in diameter, must be placed so as to just hide the moon, the
angular diameter of the moon, that is the angle its diameter subtends at
the observer's eye, being taken to be 30'.
12. A person walks in a straight line toward a very distant object and
observes that at three points A, B, and C the angles of elevation of the
top of the object are a, 2a, and 3a respectively; prove that
AB = 3BC nearly.
13. If 0 be the number of radians in an angle which is less than
a right angle, prove that
02 04
cos 0 is <1 - +0-2.
2 + 16'
14. Prove the theorem of Euler, viz. that
0 0 0
sin 0=0. cos. cos -. cos....ad. inf.
0 0 22 sin 0 0 0
We have sin0 = 2 sin cos = 22 sin cos2 cos
2 2 22 2
0 0 0 0
= 23 sin 3 cos cos cos......
0 0 0 0 0
= 2 sin yx x cos. cos c. cos c...... os 2
2n 2' 22n 2.
Make? indefinitely great so that, by Art. 228 Cor.,
0
2n sin -= 0.
0 0 0 -1
Hence sin 0=. cos. cos cos......adinf.
2 2" 2" ad j
15. Prove that
(1 tan2) (-an2)- - tan2.3).a inf.
=0. cot0.