25. What is the angle at the centre of a circle if the corresponding chord is equal to of the radius?

26. Find the area of a circular sector if the radius of the circle=12, and the angle of the sector=30°.

§ 16. THE REGULAR POLYGON.

Lines drawn from the centre of a regular polygon (Fig. 19) to the vertices are radii of the circumscribed circle; and lines drawn from the centre to the middle points of the sides are radii of the inscribed circle. These lines divide the polygon into equal right triangles. Therefore, a regular polygon is determined by a right triangle whose sides are the radius of the circumscribed circle, the radius of the inscribed circle, and half of one side of the polygon.

If the polygon has n sides, the angle of this right triangle at the centre is equal to

If, also, a side of the polygon, or one of the above-mentioned radii, is given, this triangle may be solved, and the solution gives the unknown parts of the polygon.

8. Find the side of a regular decagon inscribed in a unit circle.

9. Find the side of a regular decagon circumscribed about a unit circle.

10. If the side of an inscribed regular hexagon is equal to 1, find the side of an inscribed regular dodecagon.

11. Given n and c, and let b denote the side of the inscribed regular polygon having 2n sides; find b in terms of n and c.

12. Compute the difference between the areas of a regular octagon and a regular nonagon if the perimeter of each is 16.

13. Compute the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 12.

14. From a square whose side is equal to 1 the corners are cut away so that a regular octagon is left. Find the area of this octagon.

15. Find the area of a regular pentagon if its diagonals are each equal to 12.

16. The area of an inscribed regular pentagon is 33 8; find the area of a regular polygon of 11 sides inscribed in the same circle.

17. The perimeter of an equilateral triangle is 20; find the area of the inscribed circle.

18. The area of a regular polygon of 16 sides, inscribed in a circle, is 100; find the area of a regular polygon of 15 sides, inscribed in the same circle.

19. A regular dodecagon is circumscribed about a circle, the circumference of which is equal to 1; find the perimeter of the dodecagon.

20. The area of a regular polygon of 25 sides is equal to 40; find the area of the ring comprised between the circumferences of the inscribed and the circumscribed circles.

CHAPTER III.

GONIOMETRY.

§ 17. DEFINITION OF GONIOMETRY.

In order to prepare the way for the solution of an oblique triangle, we now proceed to extend the definitions of the trigonometric functions to angles of all magnitudes, and to deduce certain useful relations of the functions of different angles.

That branch of Trigonometry which treats of trigonometric functions in general, and of their relations, is called Goniometry.

§ 18. ANGLES OF ANY MAGNITUDE.

Let the radius OP of a circle (Fig. 20) generate an angle by turning about the centre O. This angle will be measured by the arc described by the point P; and it may have any magnitude, because the arc described by P may have any magnitude.

A'

B

B'

FIG. 20.

P

Let the horizontal line ОA be A the initial position of OP, and let OP revolve in the direction shown by the arrow, or opposite to the way clock hands revolve. Let, also, the four quadrants into. which the circle is divided by the horizontal and vertical diameters

AA', BB', be numbered I., II., III., IV., in the direction of the motion.

During one revolution OP will form with OA all angles from 0° to 360°. Any particular angle is said to be an angle of the quadrant in which OP lies; so that,

Angles between 0° and 90° are angles of Quadrant I. Angles between 90° and 180° are angles of Quadrant II. Angles between 180° and 270° are angles of Quadrant III. Angles between 270° and 360° are angles of Quadrant IV. If OP make another revolution, it will describe all angles from 360° to 720°, and so on.

If OP, instead of making another revolution in the direc tion of the arrow, be supposed to revolve backwards about 0, this backward motion tends to undo, or cancel, the original forward motion. Hence, the angle thus generated must be regarded as a negative angle; and this negative angle may, obviously, have any magnitude. Thus we arrive at the conception of an angle of any magnitude, positive or negative.

§ 19.

GENERAL DEFINITIONS OF THE FUNCTIONS.

The definitions of the trigonometric functions may be extended to all angles, by making the functions of any angle equal to the line values in a unit circle drawn for the angle in question, as explained in § 4. But the lines that represent the sine, cosine, tangent, and cotangent must be regarded as negative, if they are opposite in direction to the lines that represent the corresponding functions of an angle in the first quadrant; and the lines that represent the secant and cosecant must be regarded as negative, if they are opposite in direction to the moving radius.

Figs. 21-24 show the functions drawn for an angle AOP in each quadrant, taken in order. In constructing them, it must be remembered that the tangents to the circle are always drawn through A and B, never through A' or B'.

Let the angle AOP be denoted by a; then, in each figure,