## 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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**tangent**of 45 degrees is radius . For then the angle ACT being**half**a right angle , the other acute angle ATC must also be**half**a right angle ; therefore AC = AT ( 6. 1 ) . 29. The secant of 0 ( or at the beginning of the circle ) is ... Page 17

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**tangent of half**the sum of the opposite angles is to the**tangent of half**their difference . Let ABC be the proposed triangle , whose sides are AC , BC , and base AB . About the centre C , with the radius CB , the less of the two ... Page 18

... tangent of the angle EBF , the semi - difference of the angles at the base . The triangles AFE , AHB , having ... half the sum of the angles at the base :

... tangent of the angle EBF , the semi - difference of the angles at the base . The triangles AFE , AHB , having ... half the sum of the angles at the base :

**tan . of half**their difference . 62. Scholium . If two sides AC , BC of any ... Page 24

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**tangent of 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 ... Page 54

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**half**as far from it ? 6. What is the altitude of the sun when a man's shadow is**half**his height ? and what is the altitude of the sun when a man's shadow is**double**his height ? 1 : 2 :: R :**tan**. sun's altitude . Answer . 63 ° 26 ' . 2 ...### 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.