Plane trigonometry with practical applications, by Leonard E. Dickson.

160

TRIGONOMETRY

[Art. 110

3. Copy Figs. 86 and 87, but label in terms of radians each angle
which is a multiple of 90~.
4. Using Fig. 89, sketch the cotangent curve for angles from -27r
to + 3w.
5. Find the number of radians in an angle at the center of a circle of
radius 50 feet which intercepts an arc of 75 feet.
6. Find the length of an arc subtending an angle of 2.5 radians at the
center of a circle of radius 50 feet.
7. Find the length of the radius of a circle at whose center an angle
of 2.1 radians is subtended by an arc 42 feet long.
8. Find the length of an arc of 70~ on a circle of 9 ft. radius, using
7r = 22/7.
9. If the angle of a sector of a circle of radius r contains n radians,
the area of the sector is equal to the product ~nr2 of n / (2 r) by the area
r r2 of the circle. Subtracting the area 2r2 sin n of the triangle two of
whose sides are radii and included angle is n (Art. 86), we obtain I (n -
sin n) r2 as the area of the segment of the circle. Compute the area of a
circular segment of radius 10 feet whose arc is 40~.
3w 5w w 7w
10. Prove that cos + cos ~ +2 cos -7 cos -= 0.
11 11 11 11
Express in terms of radians all solutions of the following equations:
11. cos x + tan x = sec x. 12. cot x - csc 2 x = 1.
13. tan 2 x tan 3 x =1.
14. In field artillery, a mil is the angle subtended by an arc equal
to 1/6400 of the circumference. Show that a mil is approximately onethousandth of a radian.
15.* An endless band passes around two wheels the diameter of one
of which equals the circumference of the other. When the larger wheel
makes one complete revolution, what is the number of radians in the
angle described by the radius of the smaller wheel?
16.* Two straight railroad tracks intersect at P making an angle of
55~. They are to be connected by a curved track, tangent to each, and
forming a circular arc of radius 300 feet. Find the length of the curved
track and the distance from P to the point of tangency.
17.* Three circles, whose radii are proportional to 1, 3, 5, are tangent
externally. The points of tangency are the vertices of a curvilinear triangle whose area is 10 square inches. Find the radii.