JUNE 1894. 1. Express (a) an angle of 2 radians in degrees; (b) an angle of 30° in radians. 2. Give simple equivalents for the following functions: tan (x), cosec (x), sin(x+2), sin (x - 1⁄2π), tan (11⁄2 x), sin (27 — X). 3. Given tan x (a ÷ b), to express sin x, cos x, cot x, sec x, and cosec x in terms of a and b. 6. Given 180° << 270°, and log cot d= 0.3232, to determine d. 7. The sides of a triangle are a 32.5 ft., b = 33.1 ft., c = 32.4 ft. Calculate the area of the triangle and the angle C opposite the side c, using the following formulae: Sp(pa) (p − b)(p — c) = 1⁄2ab sin C, in which S denotes the area of the triangle, and p = 1⁄2 (a +b+c). SEPTEMBER 1894. 1. Express (a) the angle (754) in degrees; (b) the angle 22° 30′ in radians. 2. Express cos 160°, sin 215°, tan 285°, cot 475°, each as a function of a positive angle less than 90°. 3. Find a formula to express all angles which have the same tangent as the given angle A. 4. Express tan d in terms of each of the other simple trigonometrical functions. 5. (a) Given tan (a + b): 34, tan a 1⁄2, to find tan b. 2 tan 20 (b) Derive the formula tan 6 = I tan2 26 (c) Assume the preceding formula and calculate tan 1⁄2 in terms of tan 6. Also determine the proper sign to give to the radical in the result when o < < 1⁄2π, and when Υπ < δ < π. 6. Given o° << 180° and log cos' λ == A 1.82110 to find λ. 7. Two sides of a triangle are a 201.2 ft., b ft., and the angle A, opposite the side a, is 125°; solve the triangle. 155.4 JUNE 1895. 1. Express an angle of 25°.7 in radians. 2. Given sin 0 = , to find the other trigonometrical functions of 0. 6. Explain the method of solving a triangle ABC, when two sides a, b, and the included angle C, are given. Use of Logarithms. 1. Find the logarithms of the following numbers: 1. (6.608); 2. 0.6608. 2. Find the number whose logarithm is 2.83000. 3. Calculate the value of 23 X 75 V 13 X 0.85 with logarithms. 4. Find (a) the logarithmic cosine of 37° 10′.5; (b) the smallest angle whose logarithmic cotangent is 0.04220. SEPTEMBER 1895. 1. Express an angle of 1.5 radians in degrees. sin' y 4. Prove that cos (x + y) cos (x − y) = cos2 x = cos2 y — sin2 y. 5. Given tan x = 1⁄2 and tan y 3, to find tan (x—y). 6. Write the formulae for solving a triangle ABC, when two sides a, b, and the angle A (opposite a) are given. Explain how you determine the number of solutions. Use of Logarithms. 1. Find the logarithms of (1) (2.755)3, (2) 10.2755. 2. Find the number whose logarithm is 8.43016 IO. 4. Find the log cot and log sin of 39° 9′.8. 5. Find the smallest angle whose log cos is 9.51209—10. JUNE 1896. 1. Express an angle of 0.5 radians in degrees. 2. Write the fundimental relations between the six simple trigonometric functions. 3. Deduce the simplest equivalents for the sin, tan, cos, and cot of (π + α). 6. Deduce a formula for finding the cosine of an angle, A, of a triangle, in terms of the three sides, a, b, and c. 4. Find (a) the logarithmic cotangent of 45° 20′.4; (b) the smallest angle whose logarithmic cotangent is 0.05304. |