8. If the circumference of a circle be 30 in., find the arcs opposite, 30°, 3′, 20o. 9. How many radians and how many degrees are subtended by: 2 radius-arcs, radius-arcs, 3 quadrants? 10. How many radians in 17° 13' 15", 10°, 2009? 11. Convert into grades, minutes, and seconds: 1', 86° 13′ 17′′. 13. An angle of three radians at the centre of a sphere subtends a two-foot arc of a great circle; find the radius. § 6. ADDITION OF ANGLES. Two or more angles that lie in the same plane are added by placing the initial line of the second angle upon the terminal of the first, the initial of the third upon the terminal of the second, and so on; and the sum of all the angles so added is the angle bounded by the first initial and the last terminal line. When a positive angle is added, the terminal line swings forward; when a negative angle is added, it swings backward. And this is true whether the angles be positive or negative. E.g., XOPPOQ is always one of the congruent angles xoq. So, if l, m, n be any three lines in a plane, and ... + 4 rs = ls <lm+mn + ··· + < rs + 2 sl = ≤ ll = 2nπ, ... wherein n may be 0 or any other integer, positive or negative. To subtract an angle is to add its opposite. § 7. COMPLEMENT AND SUPPLEMENT OF AN ANGLE. The complement of an angle is its defect from a right angle; i.e., it is the remainder when from a right angle the given angle is subtracted. The supplement of an angle is its defect from two right angles; i.e., it is the remainder when from two right angles the given angle is subtracted. The complement of a positive angle less than a right angle is a positive angle less than a right angle; of a positive angle greater than a right angle, a negative angle; of a negative angle, a positive angle greater than a right angle. So, the supplement of a positive angle less than two right angles is a positive angle less than two right angles; of a posi tive angle greater than two right angles, a negative angle; of a negative angle, a positive angle greater than two right angles. In the figure below, xor is a right angle, xox' two right angles, xop any angle, PoY its complement, Pox' its supplement. 1. Find the complements and the supplements of the angles: 39°, 215°, 107° 12' 15", 36° 12',,,, . 2. Write down the complement of, and the supplement of , in radians, degrees, and grades. 3. Find the angle, the supplement of whose complement is 150°. § 8. THE BEARING AND DISTANCE OF A POINT. If ox be a given line in a plane and o a given point upon it; then any other point P lying in the plane is determined by joining op, and noting the size and sign of the angle xOP and the length and sign of the line op. XOP is the vectorial angle or bearing, and Op is the radius vector or distance of the point P, as to the axis ox and the origin o. The bearing XOP and the distance op, taken together, are the polar coordinates of the point P. But the position op may be reached by turning through either of the positive or either of the negative angles xop. The point P may also be determined by measuring either of the positive or either of the negative angles xOP' and the negative distance OP; so that for every point there are four pairs of polar coordinates wherein the vectorial angle lies between -360° and +360° : +xop, top; ̄xOP, top; ≤+XOP', OP; XOP', ̄OP. E.g., the same point is determined by the coordinates: +60°, +10; 300°, +10; +240°, -10; 120°, -10. EXAMPLES. Construct the points whose polar coordinates are: 30°, 10; -330°, 10; 30°, -10; 210°, 10; -150°, 10; -330°, -10; π, 8; π, −8; ́π, 8; ̄π, ̄8; 0°, 8; 0°, −8; 흉ᅲ, 4; 흉ᅲ, -4; ᅲ, 4; ᅲ, -4; 뭏ᅲ, 4; ᅲ, -4; π, -4. II. TRIGONOMETRIC FUNCTIONS. § 1. FUNCTIONS OF A SINGLE ANGLE. Let XOP be any angle, positive or negative, P any point upon the terminal line, and AP ordinate of P as to ox; then op, OA, AP are the distance, abscissa, and ordinate of P; and the six ratios of these three lines are the six principal trigonometric functions of the angle xop, viz. : CSC XOP. secant of xOP sine are defined by the equa- distance: ordinate r: y The versed sine and coversed tions: vers XOP = 1 — cos XOP, The expressions sin-1a, cos-1a, tan-1a, are anti-functions, and are read the anti-sine of a, the anti-cosine of a, ...; they mean the angle whose sine is a, the angle whose cosine is a, E.g., if a = sin 0, if b = cos 0, ... .... |