6. Find the perimeter of a regular pentagon circumscribed about a circle whose radius is 18.24 feet. 7. Find the perimeter of a regular polygon of n sides circumscribed about a circle whose radius is r. 8. Find the length of a chord subtending a central angle of 63° 14′ 20′′ in a circle whose radius is 124.93 feet. 9. A straight road, PR, makes an angle of 19° 27' 30" with another straight road, PS. Having given PR 640 feet, find RS, the perpendicular distance from R to PS. = 10. At a certain point the angle of elevation of a mountain is 34° 28'. At a second point 500 feet farther away, the angle of elevation is 31° 12'. Find the height of the mountain above the table-land. 11. One side of the square base of a right pyramid is 15 inches, and the altitude is 20 inches. Find the slant height and the edge. Find the inclination of a face to the base of the pyramid. 12. How far can you see from an elevation of 2000 feet, assuming the earth to be a sphere with a radius of 3960 miles? 13. Two forces of 95.75 pounds and 120.25 pounds, at right angles to each other, act at a point; find the magnitude of their resultant and the angle it makes with the greater force. 14. Find the velocity of a point, whose latitude is 44° 30' 20", due to the rotation of the earth. Assume the radius of the earth to be 3960 miles. 15. Find the length of a belt running around two pulleys whose radii are 12 inches and 4 inches, respectively, the distance between the centers of the pulleys being 6 feet. 16. A diagonal of a cube and a diagonal of a face of a 17. A force of 2000 pounds applied at the origin makes an angle of 33° 25' with the positive X-axis. Find its components along the X and Y axes respectively. 18. A force of 185 pounds applied at the origin makes an angle of 82° 12' with the positive X-axis. Another force of 327 pounds applied at the origin makes an angle of 11° 32' with the positive X-axis. Find the X and Y components of each of these forces. Add the X-components and also the Y-components and then find the resultant in magnitude and direction. 19. A cylindrical tank, whose axis is horizontal, is 8 feet in diameter and 12 feet long. The tank is partly filled with water, so that the depth of the water at the deepest point is 3 feet. How many gallons of water are in the tank, there being 7 gallons in a cubic foot? 20. A man who can paddle his canoe 5 miles per hour in still water paddles at his usual rate directly across a river one-half mile wide. If the river flows 4 miles per hour, where will the canoe land and what is its speed? CHAPTER IV VARIATIONS OF THE TRIGONOMETRIC FUNCTIONS REDUCTION OF FUNCTIONS OF n 90° ± a 39. In the study of the variations of the trigonometric functions, their values at 0°, 90°, 180°, 270°, and 360° are of special importance. Values of functions of 0°. The terminal line of 0°` coincides with the positive X-axis. For a point on this line, at a distance r from the origin, we have Values of functions of 90°. The terminal line of 90° coincides with the positive Y-axis, therefore 0 8 0 Values of functions of 180°. The terminal line of 180° co incides with the negative X-axis, Values of functions of 270°. The terminal line of 270° coincides with the negative Y-axis, therefore The values of the functions of 360° are identical with the values of the functions of 0°, since these angles are coterminal. 40. Variation of the functions. It has been shown that the trigonometric functions are functions of an angle. As the angle varies, the trigonometric functions depending upon the angle also vary. traced by observing the changes in the abscissa and ordinate of the point P. a. As the angle a increases from 0° to 90°, х 818 r У .।४ ४। y r х y y increases from 0 to r, and a decreases from r to 0. is positive and increases from 0 to 1 is positive and decreases from 1 to 0 is positive and increases from 0 to ∞ is positive and decreases from 0 to 0 is positive and increases from 1 to ∞ is positive and decreases from ∞ to 1 As the angle a increases through 90°, x passes through zero, changing from a positive number to a negative number. Then, immediately before a becomes 90°, cos a or is a very small positive number; while immediately after a has passed 90°, cos a is a very small negative number. This may be expressed by saying that cos a passes through |