The theorem shows that the trigonometric functions are periodic, for if is increased by 360° the values of the functions are unchanged. It will appear later that 360° is the period of the sine and cosine, and their reciprocals, while 180° is the period of the tangent and cotangent. This is a characteristic property of the trigonometric functions, which distinguishes them from all algebraic functions. EXERCISES 1. Find all the functions of each of the so-called quadrantal angles: (a) 0°; (b) 90°; (c) 180°; (d) 270°; (e) 360°. 2. Find all the functions of the angles: 3. What positive angles less than 360° have the same functions as (a) 540°, (b) – 60°, (c) 1320°, (d) — 675°, (e) 653°? 4. Determine the sign of: (a) cos 0, 0 in quadrant II. (b) tan 0, 0 in quadrant III. 5. Determine the sign of each of the functions in each of the quadrants. Tabulate the results. 6. Build a table of values of sin 0 for 0 = 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 315°, 300°, 330°, 360°. with center 0 and radius 5, and draw the line parallel to the x-axis and 3 units below it. Let their intersections be P and P'. Then the lines OP and OP' are the terminal lines of the required angles. How many such angles are there? How many positive and less than 360°? 9. By the method of the preceding exercise, construct all the positive angles less than 360° for which 15, and 0 in quadrant II. (b) tan 0 = 2, and 0 in quadrant III. (c) sin 0 1, and 0 in quadrant IV. (d) cot 0 = 2, and 0 in quadrant I. (e) sin 0 = ‡, and 0 in quadrant II. (f) sec 0 = 13, and 0 in the fourth quadrant. 11. The intensity of light, I, varies inversely as the square of the distance, d, from the source. If a street light is at the top of a concrete post 10 feet high, AB, express I at a point C on the pavement as a function of 12. An aeroplane rises along a straight line, which makes an angle with a straight road directly below the path of the aeroplane, at the rate of 50 miles an hour. Express as a function of the speed at which an automobile must move along the road to keep under the aeroplane. 13. If the hypotenuse of a right triangle is 10, express the area as a function of one of the acute angles. 14. Construct the path of a point on the rim of a wheel rolling on a level road, by rolling a coin along a ruler, without slipping, and marking a number of positions of a point on the edge of the coin. This curve defines y, the height of the moving point above the road, as a function of x, r the distance measured along the road. Show that this function is periodic, and find its period. What can be said of the graph of a periodic function? 60. Radians. To find the number of degrees in the angle u subtended at the center of a circle of radius r by an arc whose length is r, we compare the angle with the complete angle about the center. Since angles at the center are proportional to the subtending arcs, and since an angle of 360° is subtended by the entire circumference, we have FIG. 91. Hence this angle does not depend on r, but is the same for all circles, and it may therefore be used as a unit angle. DEFINITION. A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. Equation (2) enables us to reduce radians to degrees. The most convenient form of this equation is the important relation π radians = 180 degrees. (3) From this we have, for example, that 90° π/2 radians, and it is customary to speak of π/2 radians rather than 3.1416/2 = 1.5708 radians. Similarly, It is customary to express in terms of π the number of radians in any angle which is a simple multiple or submultiple of 180°. When the degree or the right angle is used as the unit that fact is usually indicated. Thus we write ◊ = 180°, or 0 = 2 rt. 4. But if the unit is the radian we merely write = π without indicating the unit. Thus we write 0 = The number of radians in an angle is called its circular measure. It is customary to use the radian as the unit angle in drawing the graphs of the trigonometric functions (see the following section). Another elementary use angle AOB, and if arc AC r, so that AOC = 1, we have The radian is the unit used in all theoretical work in the calculus and higher mathematics. Tables for converting radians into degrees, and degrees into radians, are to be found on page 32 of Huntington's Tables. EXERCISES 1. Reduce the following angles to radians: (a) The quadrantal angles, namely, 0°, 90°, 180°, 270°, 360°. (b) The angles whose terminal lines bisect the quadrants, namely, 45°, 135°, 225°, 315°. (c) The angles whose terminal lines trisect the quadrants, namely, 30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°. 2. Reduce the following angles to degrees: (а) π/6, 3π, 7π/2, 2π/3. (b) 2π/3, π/4, 5π/3, 5π/4, 11π/3. 3. Using Huntington's Tables, page 32, express the following angles in radians, as decimal fractions: (a) 10°, 75°, 110°, 340°, 15°.34, 3°.77. (b) The angles in Exercise 1. 4. Using Huntington's Tables, express the following angles in degrees: 0.25, 1.13, 1.465, 0.8327, 3.2476. 5. If the radius of a circle is 10 inches, find the angle subtended at the center by an arc 15 inches long. 6. If the radius of a circle is 4 inches, find the length of an arc which subtends an angle of 173° at the center. 7. A strip of tin 8 inches wide is bent into the form of a trough whose cross section is an arc of a circle. Express the angle subtended at the center as a function of the radius. 8. Find the angle at the center of the earth subtended by an arc of the equator one mile long. In doing this, which one of the radii of the earth given on the inside of the back cover of Huntington's Tables should be used? 9. If the radius of a circle is 12 inches, find the length of an arc subtending a central angle of 2π/3, and the area of the sector bounded by the arc and the radii drawn to its extremities. 10. Show that the area of a sector of a circle of radius r is r20, where O is the number of radians in the central angle of the sector. Hint. The area of a sector is one-half the product of the radius and the arc of the sector. 61. Graphs of the Trigonometric Functions. In construct ing these graphs it is customary to use the same unit on both axes, and to measure 0 in radians. = = Graph of sin 0. A sufficiently extensive table of values is obtained by taking the values of 0 for which the terminal line bounds, bisects, or trisects one of the quadrants. The values of sin for the angles 30° π/6, 45° = π/4, 60° π/3 are given in the first table in Section 57, and for the angles 0° and 210° = 7π/6 in the examples in Section 59. The table of values below is identical with that asked for in Exercise 6, following Section 59, except that the angles are now expressed in radians. To plot the pairs of values in the table, choose a convenient unit on the vertical axis, and on the 0-axis lay off = 34 units and 2π = 6 units. The points π/2 and 3π/2 are 0 obtained by bisecting these segments, giving four segments on the 0-axis corresponding to the four quadrants. The remaining abscissas may be constructed by bisecting and trisecting these segments. Having determined the points on the 0-axis corresponding to the values of 0 in the table, ordinates are erected equal to the respective values of sin 0, and the curve is then drawn. 1, Ө FIG. 93. 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4 4π/3, 3π/2, Periodicity of sin 0. If the terminal line of starts in coincidence with the positive part of the x-axis (Section 58), the angle may have any one of the values: |