## 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

Results 6-10 of 63

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**cosines**and secants of two arcs , and also the**sines**and cosecants , are re- ciprocally proportional . 43. The**sine**of any arc is equal to half the chord of double that arc . For the radius CA , perpendicular to BE , bisects the chord ... Page 11

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**sine**of that arc . 2. Again , the triangles AFB , DFB give the following analogy . AF : BF :: BF : DF . That is ...**cosine**of any arc AM , as twice the**sine**of that arc is to the**sine**of AMB double that arc . 47. The following ... Page 12

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**sine**, and the base AB the**cosine**of the angle at the base CAB . But if the base AB be made radius , the per- pendicular BC becomes the tangent , and the hypothenuse AC the secant of the angle at the base CAB . This is manifest from the ... Page 13

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**cos**. A. Whence the tangents and**sines**( and consequently the angles ) are discovered by the rule of proportion . We may invert these analogies , as follows :**Sine**C : R :: AB : AC ,**Sine**A : R :: CB : AC , Tan . A : R :: BC : AB , Tan ... Page 18

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**Sine**ZA : BC :: s . ∠ACB : AB . Note . The preceding propositions are sufficient for the re- solution of all the ...**cosine**of the angle con- tained by the two sides . Let ABC be any triangle ; 2 AB . BC : difference between AB + ...### Common terms and phrases

90 degrees adjacent angle AHDL algebra analogy angle ABC angle ACB Answer arc AC arc or angle base centre chord circle comp complement cosecant cosine cotangent Euclid's Elements find the angles find the rest formulæ 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 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.