JUNE 1889. 1. Express a circular are whose length is 2 feet, and whose radius is 5 feet, in degrees. 2. Establish the following formulae: (a) tan (- α) = tan a; (b) cos (− a) = cos a; (c) cos (π + α) cos a; (d) tan (27 - α) = tan a. 3. Write formulae expressing all values of a and B which satisfy the following equations, respectively: 4. Given tan a = (m ÷ n), to find sin a, cos a, sin 2a, cos 2a, tan 2α. 5. Show that if a and b are two sides of a triangle, and A and B the angles opposite them respectively, then 6. Compute A and B of the preceding question, supposing a 3456.7, b 3564.2, and (A + B) = 118° 10′.4. 7. Determine, with the help of the tables, the sine and cosine of 223° 10′.2. SEPTEMBER 1889. 1. Express an angle of 0.5 radians in degrees. 2. Point out in what quadrant an angle terminates: (1) When its sine is positive and its cosine negative. (2) When its sine is negative and its cosine positive. (3) When its tangent is positive and its cosine negative. 3. Which of the trigonometric functions are never numerically greater than one? Which ones are never numerically less than one? Which ones can have all values? (π 4. Write equivalent expressions for cos (1⁄2 π a); cos a); sin (27- a); tan (11⁄2π — α); sec (α - 5. Express the other trigonometric functions terms of tan a. 6. Show that sin (a + B) sin (a B) cos' cos' α. 1⁄2π). of a in sin2 a sin2 ß 7. Show that tan (+α) tan (4π a) = 2 tan 2a. 8. Given the three sides of a triangle, a = 10409, b 17087, c = 20008, to find the angles. JUNE 1890. 1. Express an angle of 10° in radians. 2. Express tan, successively, in terms of each of the other trigonometrical functions of 9. 3. Derive the formula, sin ß = 5 sin 1⁄2 cos 1⁄2ẞ; cos B 2 sin 1⁄2 5. 4. Given tan ẞm, to find tan 2. 6. Obtain a formula for expressing a side of a triangle when the remaining two sides and their included angle are given. 7. Given two angles of a triangle 68° 20.′2 and 72°, and the side opposite the first angle 2516.2 ft., to find the remaining parts of the triangle. 823632 SEPTEMBER 1890. 1. Express an angle of 0.25 radians in degrees. 2. Give the values of the different trigonometric functions of the following angles: o,, 7, 37. π 2 2 Π. 3. Derive the formula for the tangent of the difference of two arcs in terms of the tangent of the arcs. 7. The three sides of a triangle are as follows: a 2504.5, b = 2526.7, c = 2625.2; find the angles. JUNE 1891. 1. Express an angle of 18° in radians. 2. Write the simplest equivalent expressions for sin (11⁄2π + 4), cos (11⁄2 π P), tan (27 4). 3. Express cos in terms of each of the other trigonometrical functions of B. 4. Given tan 29 3 tan 9, to find 9. 6. Derive the formula sin (B + 9) = 2 sin 1⁄2 (+ 9) cos 1⁄2 ( ). 7. Given two angles of a triangle 60° 15′.2, 62a 25′.3, and the side opposite the former 1098.6 ft., to find the remaining parts of the triangle. |