267. It is important to remember that the conclusions of the foregoing articles only hold when the angle is expressed in radian measure. If any other system of measurement is used, the results will require modification.

When n is indefinitely small, is indefinitely small;

268. When is the radian measure of a very small angle, we have shewn that

Hence r tan 0=re, and therefore in the figure of Art. 265, the tangent PT is equal to the arc PA, when AOP is very small.

In Art. 270, it will be shewn that these results hold so long as is so small that its square may be neglected. When this is the case, we have

Example 1. The inclination of a railway to the horizontal plane

is 52′ 30′′, find how many feet it rises in a mile.

Let OA be the horizontal plane,

and OP a mile of the railway. Draw

PN perpendicular to OA.

Let PN=x feet,

P

N

A

then

PON=0 ;

Example 2. A pole 6 ft. long stands on the top of a tower 54 ft. high find the angle subtended by the pole at a point on the ground which is at a distance of 180 yds. from the foot of the tower.

:

On reduction, we find that the angle is 37′ 46′′ nearly.

269. If be the number of radians in an acute angle, to prove

270. From the propositions established in this chapter, it follows that if 0 is an acute angle,

Thus cos 0=1-ke2 and sin 0=0-k'03, where k and k' are

proper fractions less than and

respectively.

Hence if be so small that its square can be neglected, cos 0=1, sin 0=0.

Example. Find the approximate value of sin 10".

271.

limit of

To shew that when n is an indefinitely large integer, the

We have

sin 0=2 sin

COS

2

2

h denoting the radian measure of a small positive angle.

Also 1-cos h is positive; hence the numerator is positive, and therefore the fraction is positive;

1. A tower 44 feet high subtends an angle of 35' at a point A on the ground: find the distance of A from the tower.

2. From the top of a wall 7 ft. 4 in. high the angle of depression of an object on the ground is 24' 30": find its distanc from the wall.