## An Elementary Treatise on Plane & Spherical Trigonometry: With Their Applications to Navigation, Surveying, Heights, and Distances, and Spherical Astronomy, and Particularly Adapted to Explaining the Construction of Bowditch's Navigator, and the Nautical Almanac |

### From inside the book

Results 1-5 of 91

Page 7

A =

A = sec . B = a 10. Corollary . By inspecting the preceding equations ( 4 ) , we

perceive that the sine and cosecant of an angle are reciprocals of each other ; as

...

A =

**cotan**. B b - S = ( 4 )**cotan**. A = tang . B a sec . h A = cosec . B = 6 h 1 cosec .A = sec . B = a 10. Corollary . By inspecting the preceding equations ( 4 ) , we

perceive that the sine and cosecant of an angle are reciprocals of each other ; as

...

Page 8

A X

1 1 sec . A = or cos . A = ( 6 ) cos . A ' sec . A 1

...

A X

**cotan**. A a = 6 Х ab ab = l whence 7 cosec . A = 1 or sin . A = cosec . A sin . A1 1 sec . A = or cos . A = ( 6 ) cos . A ' sec . A 1

**cotan**. A = tan . A ' or tang . A = 1**cotan**. A As soon , then , as the sine , cosine , and tangent of an angle are known...

Page 9

Since the cotangent is the reciprocal of the tangent , we have

sin . A ( 8 ) 13. Problem . To find the cosine of an angle when its sine is known .

Solution . We have , by the Pythagorean proposition , in the right triangle ABC (

fig .

Since the cotangent is the reciprocal of the tangent , we have

**cotan**. A = cos . Asin . A ( 8 ) 13. Problem . To find the cosine of an angle when its sine is known .

Solution . We have , by the Pythagorean proposition , in the right triangle ABC (

fig .

Page 10

A ) 2 – (

) 2 = 1 ; ( cosec . A ) 2 = 1 + (

difficulty of calculating the trigonometric tables of sines and cosines , tangents

and ...

A ) 2 – (

**cotan**. A ) 2 h2 a2 h2 - 62 = 1 , a2 a2 or ( 12 ) ( cosec . A ) ? — (**cotan**. A) 2 = 1 ; ( cosec . A ) 2 = 1 + (

**cotan**. A ) . whence 16. Scholium . The wholedifficulty of calculating the trigonometric tables of sines and cosines , tangents

and ...

Page 11

... Bowditch's Navigator, and the Nautical Almanac Benjamin Peirce. Calculation

of cosine , & c . sin . A = 0.4568 9.65973 ( ar . co . ) 10.34027 cos . A = 0.8896 ( ar

. co . ) 10.05082 9.94918 tang . A = 0.5135 9.71055 ( ar . co . ) 10.28945

... Bowditch's Navigator, and the Nautical Almanac Benjamin Peirce. Calculation

of cosine , & c . sin . A = 0.4568 9.65973 ( ar . co . ) 10.34027 cos . A = 0.8896 ( ar

. co . ) 10.05082 9.94918 tang . A = 0.5135 9.71055 ( ar . co . ) 10.28945

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### Common terms and phrases

adjacent altitude apparent azimuth bearing becomes beginning Calculate called centre circle column computed Corollary corr correction corresponding cosec cosine cotan course declination departure determined diff difference difference of latitude difference of longitude dist distance earth equal equator equinox error EXAMPLES formula given gives greater Greenwich half height Hence horizon hour angle hypothenuse increase interval known latitude less logarithm longitude mean meridian method miles moon moon's motion Navigator nearly object obliquity observed obtained opposite parallax perpendicular plane polar pole position Problem proportional radius reduced right ascension right triangle Rules sailing sideral sides sine Solar eclipse Solution solve the triangle star substituted sun's Table tang tangent third transit true whence zenith

### Popular passages

Page 172 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...

Page 135 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

Page 39 - ... the following proportion : — as the sine of the angle opposite the given side is to the sine of the angle opposite the required side, so is the given side to the required side.

Page 43 - The sum of any two sides of a triangle is to their difference, as the tangent of half the sum of the angles opposite to those sides, to the tangent of half their difference.

Page 193 - The cosine of half the sum of two sides of a spherical triangle is to the cosine of half their difference as the cotangent of half the included angle is to the tangent of half the sum of the other two angles. The sine of half the sum of two sides of a spherical...

Page 40 - O will be equal to sixty degrees. Hence, if any two angles of a triangle be known, the third may be found by subtracting the sum of the two known angles from 180 degrees, the remainder will be the number of degrees in the third angle.

Page 145 - ... theorems, called, from their celebrated inventor, Napier's Rules. In these rules, the complements of the hypothenuse and the angles are used instead of the hypothenuse and the angles themselves, and the right angle is neglected. Of the five parts, then, the legs, the complement of the hypothennse, and the complements of the angles ; either part may be called the middle part.

Page 220 - ... any deviation in the plane of the instrument from the meridian, will evidently produce contrary effects upon the observed times of transit, exactly as in the upper and lower transits of the same star. The time, which elapses between the two observations, will differ from the time which should elapse by the sum of the effects of the deviation upon the two stars. In the use of this method, therefore, the time of the clock must be known, so that it can readily be reduced to sideral time. The deviations...

Page 297 - Solar Day is the interval of time between two successive transits of the sun over the same meridian ; and the hour angle of the sun is called Solar Time. This is the most natural and direct measure of time. But the intervals between the successive returns of the sun to the meridian are not exactly equal, but depend upon the variable> motion of the sun in right ascension. - The want of uniformity in the sun's motion in right ascension arises from two different causes ; one, that the sun does not move...

Page 199 - ... equator and the ecliptic, and hence, also, the position of the equinoxes. In expressing the positions of stars, referred to the vernal equinox, at any given instant, the actual position of the equinox at the instant is understood, unless otherwise stated. The right ascension of a point of the sphere is the arc of the equator intercepted between its circle of declination and the vernal equinox, and is reckoned from the vernal equinox eastward from 0° to 360°, or, in time, from 0* to 24*.