SEPTEMBER 1896. 1. Express an angle of 45°.5 in radians. 3. Deduce the simplest equivalents for the followinge xpressions: sin (270° - 6); cos (180° + B); tan (— 135°). 4. Deduce the formula sin p sin 2 cos 1⁄2 (9+0) sin (8). 5. Deduce a formula for tan 36 in terms of tan ð. 6. (a) Write the formulae for solving a triangle when two angles and a side opposite one of them are given. (b) Deduce a formula for the area of a triangle in terms of its base c, and two adjacent angles A and B. 3. Calculate the area of a triangle, having given its base c=250.75 ft. and two adjacent angles A = 55° 40′.7, B = 59° 20′.4. 4. Find (a) the log cos of 75° 20′.5; (b) the smallest angle whose log cot is o.14153. JUNE 1897. 1. Express an angle of 10° in radians to four decimals. 2. Given tan a, to find expressions for the other trigonometrical functions of p. 3. Give the simplest equivalents for sin (π + α); cos (1⁄2 +α); tan (11⁄2π + α); est (2π — α). 4. Derive the formula sin 3α = 3 sin a 4 sin3 a. 5. Derive the formula sin p+ sin q 2 sin (p + q) cos 1⁄2 (p − q). 6. Derive a formula for the area of a triangle, having given two of its sides, a and b, and the included angle C. Use of Logarithms. 1. Calculate the value of x = 2. 0.525 X 0.054 351 X 0.062 Calculate the area of a triangle having two sides respectively 5012 and 5200 feet, and the angle included between them 59° 20′.3. 3. Find (a) the log cos of 55° 20′.6; (b) the smallest angle whose log cos is 9.72750 – IO. SEPTEMBER 1897. 1. Express an angle of 1.5 radians in degrees. 2. Give the values of the different trigonometric functions of o, 1⁄2π, π, 11⁄2 π, and 27. 3. What angles less than 47 have (a) the same sine (b) the same tangent as . 4. Given cos = a, to find expressions for the other trigonometric functions of 9 in terms of a. 5. Derive the formula tan 1⁄2α = I cos a I + cos a 6. Derive formulae for solving a triangle when two of its sides, a and b, and the included angle C are given. Use of Logarithms. 1. Calculate the value of x = 3/0.0435 X 3986 V4534 X 0.087 2. Find the smallest angle whose log cot is 9.97720 10. ACADEMIC DEPARTMENT (YALE COLLEGE) GEOMETRY JUNE 1886. (Four out of five questions are required.) 1. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles. 2. (Problem) To describe a square on a given straight line. (Show clearly in the figure the methods by which the constructions are made.) 3. An angle formed by a tangent and a chord is measured by one-half the intercepted arc. 4. If the angle C, of a triangle is equal to the sum of the angles A and B, the side AB is equal to twice the straight line joining C to the middle point of AB. 5. Find a point in a given straight line such that its distances from two given points may be equal. [Candidates offering the whole of Plane Geometry, may take three out of five of the above questions, one of the three of them being either 4 or 5, with the following: 6. Two triangles are similar when they are mutually equiangular. 7. The circumferences of two circles are to each other as their radii and their areas are as the squares of their radii.] JUNE 1887. (The candidate may omit any two propositions.) 1. To inscribe a circle in a given triangle. 2. If a perpendicular is drawn from the vertex of the right angle to the hypothenuse of a right triangle: (1st.) The two triangles thus formed are similar to each other, and to the whole triangle. (2d.) The perpendicular is a mean proportional between the segments of the hypothenuse. (3d) Each side about the right angle is a mean proportional between the hypothenuse and the adjacent segment. 3. The area of a circle is equal to half the product of its circumference by its radius. 4. Given the radius of a circle as 8 inches; find the circumference of the circle; also the area of a sector of 40° of the circle. 5. If BC is the base of an isosceles triangle, ABC, and BD is drawn perpendicular to AC, the angle DBC is equal to one-half the angle A. 6. AB is any chord and AC is tangent to a circle at A, CDE a line cutting the circumference in D and E and parallel to AB; show that the triangle ACD is similar to the triangle EAB. |