106. All the values of 0 which satisfy the equation nt 1 1 are comprised in the formula 5 77, (n) any integer; and all or none of the values of log (-a)” are comprised among those of log ( + a)", according as (n) is even or odd. 107. When a quadrilateral is capable of having a circle inscribed in it, the sums of the opposite sides are equal to one another; and if, besides, it is capable of having one circumscribed about it, its area equals the square root of the continued product of the sides. 108. Shew that the increment of the logarithmic sine of an angle varies nearly as the increment of the angle. State the exceptions. Find tab. log sin 17° Û 12", having given tab. log sin 17° ́ = 9.4663483, tab. log sin 17° = 9.4659353. 109. Express (sin 0)4m+1 and (cos 6)2m in terms of the cosines and sines of the multiples of 0. Write down the value of (sin 0)5. 110. The area of any triangle is to the area of the triangle, whose sides are respectively equal to the lines joining its angular points with the middle points of the opposite sides, as 4 to 3. 111. In the series 1 1 cot a + cosec a+cot + a cosech + coton + cosec shew that the terms beginning with the 3rd are alternately an arithmetic and geometric mean between the two preceding; be the arcs of a circle, radius measured from the same point, each of which is equal to its tangent, 22 and if ay, az prove that 1 an 112. Determine the radius of a circle which touches each of three straight lines which cut one another in terms of the sides of the triangle which they form. How many such circles can there be ? to + +6} 113. Prove that (2m-1)7 sin 2т 2m and thence resolve cos 0 into its quadratic factors. 114. Shew how to find sin l' by the continued bisection of an angle, and compute its numerical value, having given sin (60o = 212) = .00025566. :.00025566. When 0 is small, is it more advantageous to determine sin @ from the formula involving sin 20, or cos 20? 1835 115. Having given two sides and the included angle of a plane triangle, find the remaining side and angles. 116. Prove that (cos a + V - I sin a)(cos b + V I sin b)(cos c+V – I sin c) = cos (a + b + c)+ V - I sin (a + b + c): also, (cosa +v] sina) 117. Shew that if a, b, c be in geometric progression, loga N, log, N, log. N are in harmonic progression ; and that if u represent any root of the equation xn – 1 = 0, two roots of the equation x2 – 2x” cosec 2a + 1 = 0 are represented 1 n 118. Three circles whose radii are a, b, c, touch each other externally; prove that the tangents at the points of contact meet in a point, whose distance from any one of them abc 11 a + b + c 119. Resolve x:21 – 1 into its quadratic factors, and write down the result when n = 5. 43 120. If @ be a small arc, sin 0 = 0 6 nearly. What alteration must be made in this formula when is given in seconds? 121. Eliminate 6 and from the equations a (sin ? + d (cos 02 = a; a' (sin (')2 + a (cos (')2 = á'; a tan = á tan '; and shew that 1 1 1 1 + a a a a 122. If n be any whole number not divisible by 4; then will A 2T to 47 +0 2(n-1) + tan + tan + + tan + tan n n n n n n Prove it when n = 3. 123. Let a, b, c be the middle points of the sides of the triangle ABC; and S, s the sums of the squares of the three triangles whose bases are the sides of the triangles ABC, abc respectively, and common vertex any other point within or without the plane of ABC; shew that s - 4s (ABC) 4. a 124. Prove that in a plane triangle tan B = b sin C b cos C and express B in a series proceeding according to sines of the multiples of C. 125. Compare the areas of regular octagons described in and about a circle. 126. Explain the construction of a table of logarithmic 1836 sines. If it be calculated only for angles which contain degrees and minutes, shew how to find the logarithmic sine of an angle which also contains seconds. 127. Prove that cos (A + B) = cos A.cos B - sin A.sin B ; and thence find cos 105°. Write down a formula for all angles, whose tangent tan A. 128. Having given one side and the hypothenuse of a rightangled triangle, find an expression for the logarithm of the remaining side. If the given side be nearly equal to the hypothenuse, what is the most convenient formula for determining either of the angles ? 129. Find the number of degrees and minutes in an angle subtended by that circular arc which is equal to the radius. 130. The angles of a plane triangle form a geometrical progression whose common ratio is i, shew that the greatest side = 2 (perimeter). sin 12° 51' 25" . Find the sum of n terms of the series 1 + cos x + cos 2x + cos 3x + Find the probable sum of the same series when the number of terms is not exactly ascertainable, but is known to be not less than p nor greater than q. Explain the two results when n, P, q, and q - p are all infinite. 131. Shew that 2+13 2- N3 + =v2. V2+V (2+13) V2-N (2–73) Find a value of x which will render x -sin 2x + tsin 4.c to cos4 x a maximum. What form does the equation y2 Vy? – 6% = 6 tan (tot ( a m, n, i being integers. Hence write down the cube roots of 3 – 4V – 1, in the form a + BV-. 133. An object, six feet high, placed on the top of a tower, subtends an angle, whose tangent is .015, at a place whose horizontal distance from the foot of the tower is 100 feet; determine the tower's height. = 0. 134. If a, b, c... g, h be any unequal numbers, prove that 1 1 + + (a-6)(a-c)...(a,h) * (b-a)(6-c)...(6-h) 1 + (h-a)(h-b)... (h-g) Resolve cos ax sin bx into algebraic quadratic factors. 135. Express (cos 0)" in terms of the cosines of multiples of 0. Ex. (cos 0)6. 136. Upon what principle are distances, measured on a line which revolves about a fixed point in it, affected with their proper algebraic signs? Upon this principle trace the sign of the secant of an arc through 360°, and of the chord through 720o 137. Prove the formula 2 sin x = ✓ + sin 2x +vi sin 2x 2 cos X = ✓ + sin 2x sin 2x. Between what limits is a comprised ? 138. Compare numerically the areas of regular pentagons described in and about a circle ; and express the ratio in the form of a continued fraction. |