## A System of Plane and Spherical Trigonometry: To which is Added a Treatise on Logarithms |

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Page 111

... ) 2 ) — cos ( a + ( 2n + 1 ) 1⁄2 ) therefore , by adding and putting S , for the sum of the series to n terms 2 S , sin = COS ( a + 212 ) COS ( a + ( 2 n + 1 ) 2 ) 1 )

... ) 2 ) — cos ( a + ( 2n + 1 ) 1⁄2 ) therefore , by adding and putting S , for the sum of the series to n terms 2 S , sin = COS ( a + 212 ) COS ( a + ( 2 n + 1 ) 2 ) 1 )

**ηβ**sin ( 96. ) = 2 sin ( a SECTION V ON TRIGONOMETRIC SERIES. Page 112

To which is Added a Treatise on Logarithms Richard Wilson.

To which is Added a Treatise on Logarithms Richard Wilson.

**ηβ**sin ( 96. ) = 2 sin ( a a + + ( n + 1 ) 2 ) . sin ( a ( a + ( n + 1 ) ) . sin**ηβ**.. S β sin 219. COR . 1. The value of S. cannot be determined . ∞ For it is evident that the ... Page 113

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**ηβ**2 .. sin ẞ + sin 2 ß + · + sin n ẞ = sin 224. PROB . To sum the series Cos ( a + B ) + cos ( a + 2 ß ) + β 2 cos ( a + n B ) . sin 23 = sin ( a ... — + cos ( a + n ß ) . = sin ( a + ( 2n + 1 ) 2 ) — sin ( a + ( 2n − 1 ) ) Hence , by ... Page 114

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**ηβ**2**ηβ**2 a + The same result may immediately be obtained by putting π + a for a , in art . 218 . 2 225. COR . 1. For the reasons given in ( art . 219. ) it is im- possible to express the sum of this series to an infinite number of ... Page 115

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**ηβ**· sin 9/2 82 230. PROB . To sum the series sin ( a + B ) sin ( a + 2ß ) + ; + sin ( a + n ß ) according as m is even , or odd . In the formula β 2 sin ( a + nß ) cos 23 = sin ( a + ( 2 n + 1 ) 2 ) + sin ( a + ( 2n − 1 ) 1⁄2 ) by ...### Other editions - View all

### Common terms and phrases

A+B+C a+n ß a+ß a₁ B₁ base C₁ centre chord cosec cot cot diameter equal equations formula four right angles given greater or less hence hypothenuse integer intersection less than 90 Let the sides logarithm loge method Napier's rules nearly negative perpendicular plane angles plane triangle polar triangle pole PROB PROP quadrant quantity R₁ radius unity regular polyhedrons right-angled triangle Similarly sin A sin sin ß sines and cosines small circle solid angle sphere spherical angle spherical polygon spherical triangle ß₁ subtending sum the series suppose tangent trigonometric functions values vers α₁ Απ Δα ηβ π α π π

### Popular passages

Page 197 - IF two triangles have two sides of the one equal to two sides of the other, each to each ; and have likewise the angles contained by those sides equal to one another ; they shall likewise have their bases, or third sides, equal ; and the two triangles shall be equal ; and their other angles shall be equal, each to each, viz.

Page 179 - The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere.

Page 190 - The sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical triangle; produce the sides AB, AU, till they meet again in D.

Page 191 - THEOREM. The sum of the sides of a spherical polygon is less than the circumference of a great circle.

Page 181 - ... poles. 751. COR. 2. All great circles of a sphere are equal. 752. COR. 3. Every great circle bisects the sphere. For the two parts into which the sphere is divided can be .so placed that they .will coincide; otherwise there would be points on the surface unequally distant from the centre. 753. COR. 4. Two great circles bisect each other. For the intersection of their planes passes through the centre, and is, therefore, a diameter of each circle. 754.

Page 252 - A solid angle is that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point. X. ' The tenth definition is omitted for reasons given in the notes.

Page 189 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

Page 181 - An arc of a great circle may be drawn through any two points on the surface of a sphere.

Page 45 - Law of Sines — In any triangle, the sides are proportional to the sines of the opposite angles. That is, sin A = sin B...

Page 9 - The Versed Sine of an arc, is the part of the diameter intercepted between the arc and its sine.