## 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 17

Page 65

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**tangent**radius X**sine**cosine , and the cotangent = radius x cosine**sine**, and the cosecant radius2**sine**. Hence it appears from what has been said respecting the**sine**of a negative arc , that the**tangent**...**half**the sum of the sines of the ... Page 69

... cos . 1⁄2 ( A — B ) , and s . A- s . B = 2 cos . ( A + B ) × s . 1⁄2 ( A — B ) , s . A + s . B s . ( A + B ) x cos . ( A ...

... cos . 1⁄2 ( A — B ) , and s . A- s . B = 2 cos . ( A + B ) × s . 1⁄2 ( A — B ) , s . A + s . B s . ( A + B ) x cos . ( A ...

**tangent of half**the sum of the arcs : tan- gent of half their difference . If A = 90 ° , and B = 2x , then R + s ... Page 88

... CD be drawn from any angle C upon the oppo- site side , or base , AB , the

... CD be drawn from any angle C upon the oppo- site side , or base , AB , the

**sine**of the sum of the sides containing the angle C is to the**sine**of their difference , as the cotangent of**half**the sum of the segments 88 SPHERICAL**TRIGONOMETRY**. Page 89

... half the sum of the segments of the vertical angle is to the

... half the sum of the segments of the vertical angle is to the

**tangent of half**their difference . S Because tan . BC : tan . AC :: cos . ACD : cos . BCD ( 65 ) , tan . BC + tan . AC : tan . BC tan . AC :: cos . BCD + cos . ACD : cos . BCD ~ ... Page 90

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**tan**. AB :**tan**. ( BC + AC ) ::**tan**. ( BC - AC ) :**tan**. ( BD + AD ) . 71. Cor . 2. From the demon . cot . ( BC + AC ) :**tan**. ( BC AC ) :: cot . ( BD + AD ) :**tan**. ( BD AD ) ; that is , in any spherical**triangle**, the cot . of**half**...### 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.