Applying the fact proved above once more, (1) holds for all values of less than 270° and of less than 90°; etc. In like manner we may prove that may be increased by 90° and thence that (1) holds for all positive values of 0 and 4. 6. Prove that (2) holds for all positive values of 0 and 4. 7. Prove that (1) holds if is negative. Solution: No matter what the value of o may be, we can choose an integer n such that n Then sin (0+ $) = = = 360° + is positive. sin (n 360° + 0 + 0) = sin [0 + (n 360° + $)] 8. Prove that (2) holds for all negative values of 4. 9. Is it necessary to employ the methods of Exercises 5-8 to prove that (3) holds for all values of 0 and 4, positive and negative? 10. What property of (a) ", where n is a positive integer; (b) bo, is analogous to the properties of sin x, cos x, and tan x given by formulas (1), (2), and (3)? 11. If sin 1o is known, how complete a table of values of the trigonometric functions can be computed? 14. Prove that the force to make a sailboat move forward will be greatest when the direction of the sail bisects the angle between the keel and the apparent direction of the wind. 121. Functions of the Difference of Two Angles. We now seek an expression for the sine of the difference of two angles, analogous to the formula in algebra for the square of the difference of two numbers. = Since 0 0 + (-), and since the formulas in the pre0ceding section are true for all values of the angles, we have But = sin cos (-) + cos 0 sin (-). cos ( - ) = cos and sin (− ) And hence, In like manner it may be proved that and = sin . sin (0 – 6) = sin 0 cos & 122. Functions of Twice an Angle, or the Functions of Any Angle in Terms of Half the Angle. Since 2000, we have The graph of sin 20 may be obtained by bisecting the abscissas of points on the graph of sin ℗ (Theorem, page 151). The graph suggests that the period (Definition, page 168) of sin 20 is π 180°. This is, indeed, the case. For if we re = place by + 180°, we get sin 2(0 + 180°) sin (20+ 360°) sin 20. = = A general expression for the sine of the product of two numbers would be analogous to the theorem giving the log arithm of the product of two numbers (page 223), or the nth power of the product of two numbers [(3), page 153]. Formula (1) is the special case of such a theorem obtained when one of the numbers is 2. Another special case is given in equation (1) of the section following. 123. Functions of Half an Angle, or Functions of Any Angle in Terms of Functions of Twice the Angle. Since all that is essential in the formulas of the preceding section is that the angle on the left be double that on the right, formulas (2a) and (2b) may be written Solving these equations for sin 0/2 and cos 0/2 we have Dividing (1) by (2) and using the formula (4), page 333, (1) (2) (3) EXERCISES 1. Express the formulas in Sections 121, 122, and 123 in words. In Section 122, describe (a) ◊ as any angle, (b) 20 as any angle. In Section 123, describe (a) ✪ as any angle, (b) 0/2 as any angle. 2. Find all the functions of 15° from those of 45° and 30°. 3. Find the functions of π/8 from those of π/4; of π/12 from those of π/6. The table given by the Hindu Aryabhata (476 A.D. -) gives the values of the sines of angles at intervals of 3° 45'. How could this table be obtained? 4. If = 90°, show that (1), (2), (3), Section 121, reduce to formulas in Section 62, page 177. 5. If ◊ = 180°, show that (1), (2), (3), Section 121, reduce to (1), (2), (3), page 192. 6. State the properties of x3 analogous to each of the equations (1) in Sections 120, 121, 122, 123. 7. In the following, find sin ✪ and cos ✪ without finding @ given Hint: Find cos 20 from a figure (see Exercise 10, page 170) and then use formulas (1), and (2), Section 123. 8. Express sin 40 in terms of functions of 20; sin 0 in terms of functions of 0/2; sin 30 in terms of functions of 30/2. 9. Transform (3), Section 123, into the forms Then transform each of the fractions into tan (0/2) by expressing sin and cos in terms of 0/2, using the formulas in Section 122. In what respect is the latter procedure preferable? 11. A body is placed on a rough plane which is inclined at any angle greater than the angle of friction. (The angle of friction is an angle such that tan = m, the coefficient of friction.) If the body is supported by a force acting parallel to the plane, find the limits between which the force must lie. Let be the angle of inclination of the plane, W the weight of the body and R the reaction perpendicular to the plane. (a) Let the body be on the point of moving down the plane, so that the force of friction acts up the plane and is equal to mR. Let P be the force required to keep the body at rest. Resolving W into components parallel and perpendicular to the plane, we have P+mR = W sin 0, R W cos 0, and therefore P W (sin 0 - m cos 0). Since m = tan we have, P W (sin tan cos 0).. = = = (b) Let the body be on the point of motion up the plane. Complete the solution. 12. What is the maximum and the minimum force which will hold a weight of 12 pounds on a plane inclined at an angle of 40°, if the coefficient of friction is 0.5, and if the force acts parallel to the plane? 13. A block W rests on a horizontal plane. If an oblique force P acts upon W, making an angle ✪ with the direction of sliding, and if the coefficient of friction is m = tan, prove that the magnitude of P that will cause the block to slide is Find the least pull that will make the block slide. Suggestion: Show that P will be a minimum when 0 = 4, i.e., the direction of pull is given by the angle of friction. 14. If the inclination of a plane is 0 = arc sin }, what horizontal force would support a body of 50 pounds upon it? What horizontal force would be required if 0 were doubled? 15. On the same axes, plot the graphs of the functions below. Find the period of each function. (e) sin a cos (B − a) + cos a sin (ẞ − a) = sin ß. 124. Sum and Difference of the Sines or Cosines of Two Angles. To express the sum of the sines of two angles, sin a + sin ẞ, as a product, let |