10. A schooner sails 10° west of north at the rate of 7 knots an hour across the Gulf Stream at a place where it flows N.E. with a velocity of 4 knots an hour. Find the actual velocity of the schooner in direction and magnitude.

11. The current in a river flows at the rate of 2 miles an hour, and a man rows at the rate of 4 miles an hour. If he desires to cross the river at an angle of 70° with the bank, in the direction of the current, in what direction should he row?

12. Solve the preceding exercise if the man desires to cross at the same angle with the bank but in the upstream direction.

13. Resolve a velocity of 50 feet per second into two components inclined at 10° and 40° respectively to the direction of the given velocity.

14. A boy in an automobile moving 40 feet per second throws a ball in a horizontal direction inclined at 50° to the road with a speed of 30 feet per second. At what angle to the road will the ball move?

15. A road runs up a hill at an angle of 20°. At a point on it, 500 feet from the foot, the angle of depression of a horseman on the road leading to the hill is 5°. How far is he from the foot of the hill?

16. To find the width of a river, two points A and B are taken on one bank 100 feet apart. If C is a point on the opposite bank such that LBAC 62°.34 and 4ABC = 49°.82, find the width of the river.

17. At a certain point the angle of elevation of the top of a mountain is 45°, and at a point 1000 feet nearer the mountain and at the same level as the first, the angle of elevation is 54°.13. How high is the mountain?

18. A ship steams due east at the rate of 25 miles an hour, and the smoke from its funnel is blown in a direction 20° south of west. The wind gauge shows an apparent velocity of 35 miles an hour for the wind. Find the actual velocity of the wind in direction and magnitude.

74. Inverse Trigonometric Functions. To find the inverse of sin x (pages 40 and 114) we set y sin x, and interchange x and y, obtaining x = sin y. The solution of this equation for y in terms of x requires the introduction of a new function which is called the angle whose sine is x, and which is denoted by arc sin x. Hence,

A table of sines may be regarded as a table of angles whose sines are given (see "finding 0 if sin ◊ is given" page 178). Thus Example 2, page 179, might have been stated: find Ꮎ = arc sin 0.4332, the result being

arc sin 0.4332 25°.76 + n360° or 154°.33 + n360°.

This example illustrates the fact that for a given value of x arc sin x has not only one but a boundless number of values. This is apparent from the graph of arc sin x, which is sym

X

DEFINITION. The principal value of any one of the inverse trigonometric functions for a given value of x is that one of the boundless number of values of the function which is smallest numerically. If two values of the function are equal numerically, but opposite in sign, the positive value is the principal value. Unless the contrary is indicated,

the symbols arc sin x, arc cos x, etc., will be used in this work to denote the principal values only.

Thus arc sin = π/6,

and arc sin (-1)

π/2.

The part of the graph which represents the principal values of arc sin x is given in the figure.

Inverse trigonometric functions are of much importance, although

-1

we shall use them but little in this course. venient in stating a general result, as in this

2

+

arc sin x

FIG. 124.

They are con

EXAMPLE. What angle is subtended at the center of a circle of radius

r by a chord c units long?

Choose the center of the circle as the origin of a system of coördinates, and let the x-axis be perpendicular to the chord. Then the x-axis bisects the "chord and the angle formed by the

radii drawn to its extremities. From the

figure

sin 0/2 = (c/2)/r,

whence 0/2

and hence

UN

= arc sin (c/2r)

2

1. Construct the graph of arc sin x, and indicate on it the part which represents the principal values of the function for all possible values of x. State as many properties of the function as can be readily obtained from the graph.

2. Proceed as in Exercise 1 for the function (a) arc cos x. (b) arc tan x. (c) arc cot c. (d) arc secx. (e) arc csc x.

3. Find all the values of arc sin 0; arc cos ; arc tan (- 1).

4. Find the value (noting the convention with regard to principal values) of

(a) arc cos (− 1); arc tan 2, arc sin (— 0.3215).

(b) arc sin ; arc sec 2; 3 x arc sin (— √).

5. A rope l feet long is stretched from the top of a building to the ground, the lower end being d feet from the building. Find a general expression for the angle which the rope makes with the ground. What is the angle if l is 50 feet and d is 17 feet?

6. A mountain h feet high is viewed from a point d miles away (horizontally). What angle does the line from the point of observation to the peak make with the ground?

7. What is the value of sin (arc sin a)? Of arc sin (sin a)?

8. Recall the method of solution of Exercises 9 and 10, page 170. Find the value of (a) sin (arc cos ). (b) tan (arc sin ). (c) cos (arc tan − 2). (d) tan [arc cos (− 3)].

8

9. If the maximum distance from a point on an arc of a circle to the chord of the arc is d, show that the central angle subtended by the arc is 2 × arc cos (r - d)/r, where r is the radius of the circle.

MISCELLANEOUS EXERCISES

1. If a creek is 20 feet wide, and if from a point 4 feet above the water's edge on one side the angle of elevation of the top of the bank on the other side, directly opposite, is 7°.27, how high is the bank?

2. An object weighing 60 pounds is supported on a smooth plane whose inclination is 60° by a man who pushes against it horizontally. Find the force exerted by the man and the pressure on the plane.

3. A board is just strong enough to bear an object weighing 100 pounds at its middle point when the board is supported horizontally at its ends. How heavy an object will it bear at its middle point if it is supported at its ends in a position inclined at 30°, the object being held in position by a rope parallel to the plane?

4. Solve Exercise 3 if the object is held in position by a man pushing against it horizontally.

5. The hatchway into the hold of a ship is 16 feet wide. To raise an object weighing 200 pounds from the hold, two men on opposite sides of the hatchway pull on a rope which passes through a smooth ring fastened to the object. Find the tension of the rope when the ring is 6 feet below the deck. What force does each man exert? Hint: See the note following.

NOTE. The tension of a cord or rope which passes over a smooth peg, or over a pulley, or through a smooth ring is assumed to be the same on both sides of the peg, pulley, or ring. It is usually not the same if the rope is tied to the ring.

6. A cord is tied at a point A, passes through a smooth ring B weighing 3 pounds, over a pulley C at the same height as A, and to the end is tied a weight of 2 pounds. Find BAC when the ring and weight are în equilibrium.

7. A man drags a trunk across a room by pulling on the handle în a direction at 35° to the horizontal. If the trunk weighs 150 pounds, and if the friction is 0.1 of the pressure on the floor, find the force he exerts at the instant the trunk is about to move. Note that the pressure on the floor will be less than the weight of the trunk, because a part of the man's effort tends to lift the trunk.

8. Find the force exerted in pushing the trunk in Exercise 7, when the trunk is just on the point of moving, If the man pushes down on the trunk in a direction inclined at 25° to the horizontal.

9. To find the width of a river, a point A is taken on one bank directly opposite a tree on the other bank, and a point B is taken 100 feet from A in the line of the tree and A. At B the angle of elevation of the top of the tree is 32°.19, and at A it is 41°.33. Find the width of the river.

10. If a body on a rough inclined plane is just on the point of moving down the plane, show that the coefficient of friction is equal to the tangent of the angle of inclination of the plane.

11. Rain drops are falling straight down with a velocity of 20 feet per second. At what angle would they appear to fall to a man walking at the rate of 3 miles per hour? To a man in an automobile moving at the rate of 20 miles per hour? To a man in an express train moving at the rate of 60 miles an hour?

12. Prove that the area of any quadrilateral is equal to one-half the product of the diagonals and the sine of the angle between them.

13. The wind is blowing down a lake 5 miles wide at a rate which would blow a row boat a mile an hour. A man who rows at the rate of 3 miles an hour desires to go straight across. In what direction should he row,

and how long will it take him to cross?

14. Two girls hold a traveling bag weighing 40 pounds. One pulls on the handle at an angle of 10° to the vertical, the other at 15°. What force does each exert?

15. If the distance an ivory ball rolls down a smooth plane inclined at t 1, 2, 3, 4 5° in various times are as given in the table, S 1.4, 5.5, 12.5, 22.2 the units being feet and seconds, find the acceleration due to gravity. Analysis of the problem: Find s as a function of t from the table, from this find the velocity at any time, then the acceleration with which the ball rolls down the plane, and finally the acceleration due to gravity.

16. Charles' law states that the rate of increase of the volume of a gas under constant pressure per degree (Centigrade) rise of temperature is as of the volume at 0°, vo. Hence the volume v at any temperature

273

ว

is Vo +vo. Boyle's law states that the product of the pressure and 273 volume of a gas at constant temperature is constant, pv = k. If a quantity of oxygen occupies 200 cubic centimeters at a temperature of 17° Centigrade under a pressure of 742 millimeters (barometric height), find its volume at 0° temperature under a pressure of one atmosphere at sea level, or 760 millimeters. Illustrate graphically, plotting the graphs of both laws on the same axes, taking v on the vertical axis for each law. First find the volume vo under a pressure of 742 millimeters; the graph of Charles' law being determined by the given data and the fact that v = = 0 when 273°, the "absolute zero." The value of The value of k in Boyle's law is determined by the point whose coördinates are 200 and this value of v。. 17. Construct a square 8 inches on a side. Its area is 64 square inches. Cut the square into two rectangles of widths 3 inches and 5 inches. Cut the first rectangle along a diagonal into two triangles. Mark points on the 8 inch sides of the second 5 inches from opposite vertices, and cut along the line joining them. The four parts of the original square may be arranged in the form of a rectangle whose dimensions are 5 inches and 13 inches, and whose area is 65 square inches. Explain the fallacy.