The proofs hold for either figure. In deriving (2) when 0+ is obtuse, it should be noted that OB is negative, so that we will still have OB OD – BD. We shall assume that these formulas hold for all values of Ø and 4, negative as well as positive. (See Exercises 5-8 below.) Dividing (1) by (2), and using (4), page 333, we have 1. Express formulas (1), (2), (3), in words. 2. Using the functions of 30°, 45°, 60°, as given on page 161, find all the functions of (a) 75°, (b) 105°. Hint: sin 75° sin (45° + 30°), etc. 3. If arc sin and @ = = sing ø - arc cos, find the functions of 0+ p. 4. What force acting parallel to the plane is necessary to support a body weighing 100 pounds on a smooth plane if the inclination is = arc sin ? If the inclination is o - arc sin ? If the inclination is 0 + ? 5. Prove that (1) holds for all positive values of 0 and 0. Solution: We will first prove that: If (1) and (2) are true for a pair of values 0 and 6, then (1) is true when ✪ is increased by 90°. Since (1) holds for all acute values of 0 and 6, by the above it holds for all obtuse values of 0 and all acute values of 0. 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 360° + is positive. Ө Then sin (0+) = sin (n 360° + 0 + 0) = sin [0 + (n 360° + $)] sin 0 cos (n 360° + ) + cos 0 sin (n 360° + ) sin cos + cos 0 sin p. 8. Prove that (2) holds for all negative values of 0. 9. Is it necessary to employ the methods of Exercises 5-8 to prove that (3) holds for all values of 0 and 9, positive and negative? 10. What property of (a) x", where n is a positive integer; (b) b2, is analogous to the properties of sin x, cos x, and tan x given by formulas (1), (2), and (3)? 11. If sin 1° is known, how complete a table of values of the trigonometric functions can be computed? (c) Sin (x + y) cos y + cos (x + y) sin y = sin (x + 2y). 13. Solve the equations: 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 preceding section are true for all values of the angles, we have sin (0 – 6) = sin (0 + (− Ø)) sin 0 cos (-) + cos 0 sin (- Ø). 122. Functions of Twice 1+tan 0 tan * 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 0 (Theorem, page 151). The graph suggests that the period (Definition, page 168) of sin 20 is 180°. This is, indeed, the case. For if we re place T = by 0 + 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) 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. 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°. In Section |