## 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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**half**a right angle , the other acute angle ATC must also be**half**a right angle ; therefore AC = AT ( 6.1 ) . 29. The ...**sum**or the difference of the cosine and radius , namely , the**sum**in the two middle quadrants HD and DL , and the ... Page 16

... half the difference added to

... half the difference added to

**half the sum**; and BC = DC - DB , that is , to the excess of**half the sum**above half the difference . 58. Cor . If the semi - sum be subtracted from the greater quantity , the remainder will be the semi ... Page 17

... sum of any two sides is to their difference , as the tangent of

... sum of any two sides is to their difference , as the tangent of

**half the sum of**the opposite angles is to the tangent of half their difference . F H Let ABC be the proposed triangle , whose sides are AC , BC , and base AB . About the ... Page 18

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**half the sum of**the angles at the base : tan . of half their difference . 62. Scholium . If two sides AC , BC of any triangle ABC , and the angle ACB included between them , be given , the other side and angles may be found . For , from ... Page 24

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**half the sum of**the two unknown angles we may take the cotangent of half the given angle , or the tangent of half its supplement ; for these three tangents are equal to one another . Thus , the sum of the angles B and C is less than two ...### 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.