## A Treatise of Plane and Spherical Trigonometry: In Theory and Practice ; Adapted to the Use of Students ; Extracted Mostly from Similar Works of Ludlam, Playfair, Vince, and Bonnycastle |

### From inside the book

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**sines**,**tangents**, secants , and versed sines , computed for every degree and minute of the first quadrant , will ... tangents and cotangents of any two arcs of a circle are reciprocally proportional ; that is , the tangent of the first ... Page 13

... sine C. Whence the sine of the angle C is known , and consequently the quantity of the angle C may be found by seeking the sine in a table of

... sine C. Whence the sine of the angle C is known , and consequently the quantity of the angle C may be found by seeking the sine in a table of

**sines and tangents**. Again , AC : CB :: R : sine A , and AB : BC :: R : tan . A , and BC : AB :: R ... Page 27

... sine of BAC is the same as the sine of B + C . The angle B Ac AcC — B ( 32. 1 ) = ACc - B . Hence , the angle C being ...

... sine of BAC is the same as the sine of B + C . The angle B Ac AcC — B ( 32. 1 ) = ACc - B . Hence , the angle C being ...

**sines and tangents**. The construction PLANE TRIGONOMETRY . 27 Observations preparatory to the Practical Solution ... Page 28

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**sines**,**tangents**, & c . are also called the natural**sines**,**tangents**, & c . of the arcs or angles to which they belong ; and the logarithms of the numbers by which they are represented , are called the logarithmic**sines**,**tangents**... Page 33

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**tan**. A ÷ R , and AC = AB × R ÷ cos . A = ABX R ÷**sine**C = AB x sec . A ÷ R . Solution by Natural**Tangents**and Secants . * 1 : 1 · 2489484 :: 125 : BC = 156 · 11855 . 1 : 1.5964824 :: 125 : AC = 199.5603 . Solution by Logarithmic**Sines**, ...### Common terms and phrases

90 degrees adjacent angle AHDL algebra analogy angle ABC angle ACB Answer arc or angle base centre chord circle comp complement cosecant cosine cotangent Euclid's Elements find the angles find the rest geometry Given the side greater than 90 half the sum half their difference height Hence hypothenuse AC included angle less than 90 logarithmic sines mathematics measured mechanical philosophy negative opposite angle perp perpendicular plane triangle plane trigonometry PROP propositions quadrant AH quantity right-angled spherical triangle right-angled triangle Scholium secant side AB side AC sides and angles sine a sine sine and cosine sine² sines and tangents solution spherical angle spherical triangle ABC spherical trigonometry supplement tables tangent of half theorems third side three angles three sides triangle are given trigono versed sine yards

### Popular passages

Page 12 - In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference.

Page ix - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.

Page 23 - Then multiply the second and third terms together, and divide the product by the first term: the quotient will be the fourth term, or answer.

Page 13 - In any triangle, twice the rectangle contained by any two sides is to the difference between the sum of the squares of those sides, and the square of the base, as the radius to the cosine of the angle included by the two sides. Let ABC be any triangle, 2AB.BC is to the difference between AB2+BC2 and AC2 as radius to cos.

Page 87 - 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 74 - The sum of any two sides is greater than the third side, and their difference is less than the third side.