Elements of Geometry, and Plane and Spherical Trigonometry

14. The equatorial diameter of the earth is 7925 miles, and the distance of the sun 91,500,000 miles. What angle will the semi-diameter of the earth subtend, as seen from the sun? Ans. 8.94".

This angle is called, in astronomy, the sun's horizontal parallax. The preceding figure applies to this example, by supposing E to be the center of the sun, S that of the earth, and BS equal to 3956 miles.

15. The mean distance of the moon from the earth is 60.3 times 3960 miles, and at this distance the semidiameter of the moon subtends an angle of 15′ 32". What is the diameter of the moon in miles?

Ans. 2157.8 miles.

II. OBLIQUE-ANGLED TRIGONOMETRY.

PROBLEM I.

In a plane triangle, given a side and the two adjacent angles, to find the other parts.

In the triangle ABC, let AB

376, the angle A 48° 3', and the angle B 40° 14', to find the other parts.

As the sum of the three angles of every

triangle is always 180°, the third angle, C, must be 180° 88°

17' 91° 43'.

Observe, that the sine of 91° 43′ is the same as the cosine of

1° 43'

In a plane triangle, given two sides and an angle opposite one of them, to determine the other parts.

Let AD 1751 feet, one

of the given sides; the angle D= 31° 17' 19"; and the side opposite, 1257.5. From these data, we are required to find the other side and the other two angles.

In this case we do not know whether AC or AE represents 1257.5, because AC = AE. If we take AC for the other given side, then DC is the other required side, and DAC is the vertical angle. If we take AE for the other given side, then DE is the required side, and DAE is the vertical angle. In such cases we determine both triangles.

From 180° take 46° 18′, and the remainder is the angle DCA

The angles D and E, taken from 180°, give

DAE 102° 24′ 41′′.

REMARK.—Tɔ make the triangle possible, AC must not be less than AB the sine of the angle D, when DA is made radius.

PROBLEM III.

In any plane triangle, given two sides and the included angle, to find the other parts.

Let AD = 1751, (see last figure), DE 2364.5, and the included angle D 31° 17' 19". We are required to find AE, the angle DAE, and the angle E.

Observe that the angle E must be less than the angle DAE, because it is opposite a less side.

DE + DA: DE — DA = tan. 74° 21′ 20′′ : tan. }(DAE—E)

That is,

4115.5 : 613.5 = tan. 74° 21′ 20′′ : tar. (DAE — E)

tan. (DAE-E) tan.28° 1' 36", 9.726170

But the half sum plus the half difference of any two quantities is equal to the greater of the two; and the half sum minus the half difference is equal the less.