### Popular passages

Page 35 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the 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. By Theorem II. we have a : b : : sin. A : sin. B.
Page 61 - The law of sines states that in any spherical triangle the sines of the sides are proportional to the sines of their opposite angles: sin a _ sin b __ sin c _ sin A sin B sin C...
Page 48 - From half the sum of the sides subtract each side separately. Multiply the half sum and the several remainders together, and the square root of the product will be the area.
Page 60 - ABC ; then'.' cos A = cos a sin B, and sin B is positive, .'. cos A, cos a are positive, negative, or zero together, .'. A, a are both acute, both obtuse, or both right. QED THEOR. 21. In an ideal right triangle, if the hypotenuse be acute the two oblique sides are of the same species, and so are the two oblique angles ; but they are of opposite species if the hypotenuse be obtuse. For let c be the right angle in the triangle ABC ; then'.
Page 34 - Law of Sines — In any triangle, the sides are proportional to the sines of the opposite angles. That is, sin A = sin B...
Page 53 - Two observers on the same side of a balloon, and in the same vertical plane with it, are a mile apart, and find the angles of elevation to be 17� and 68� 25' respectively : what is its height ? [1836 feet.