173. We may also conversely prove that the 'A, B' formulæ are true for angles of any magnitude by the aid of the result of page 107. (This is a very convenient method of proving the 'A, B' formulæ to be true for all values of the angles.) For, assuming that the ' A, B' formulæ are true for certain values of the angles A and B, we can show that they are true if either of the angles A or B be increased by 90o. Example. sin (90°+A+B) = =cos (A + B) [p. 107.] =cos A. cos B - sin A. sin B, =sin (90°+A). cos B-{-cos (90°+A)} sin B, =sin (90°+A). cos B÷cos (90°+4). sin B. Or, writing A' for 90o+A, we have sin (A+B)=sin A'. cos B+cos A'. sin B. We have proved (Art. 170) that the 'A, B' formulæ are true for all values of A and B between 0° and 90o. And therefore, by what we have said above, they are true for all values of A or B between 0° and 180o. And so on. Therefore the 'A, B' formulæ are true for any values whatever of the angles A and B. 174. It follows that all formulæ deduced from the 'A, B' formulæ are true for angles of any magnitude what ever. Thus the S, T' formulæ (page 126) are true for angles of any magnitude. Also the formula of the last Chapter for multiple angles, and all general formulæ in the Examples, are true for angles of any magnitude. EXAMPLES. XLV. (1) Deduce the values of sin 180° and cos 180° from those of the sine and cosine of 90o. (2) The angle A is greater than 180° and less than 270° and tan A=: find sin 24 and sin 3A. lies in the fourth quadrant and cos 0=1, (3) The angle find sin 20 and sin 30. which quadrant 30 lies. Find also cos 30, and hence determine in (4) Prove that the different values of which satisfy the equation cos pe+cos q0=0, form two series in A. P. with com 175. We have now proved all the formulæ of the last two Chapters to be universally true. We may expect therefore that any result, which can be obtained from these formula by algebraical transformation, will have a complete geometrical interpretation. [See p. 116.] Thus, given the value of cos A (nothing else being known about the angle 4), we get two values for sin A 2' tive and one negative, and two like values for cos A 2 one posi 177. To prove geometrically that, given the value of cos A (nothing else being known about the angle ▲), there Let a be the least positive angle which has the given cosine, and let ROP, and P,OR in the figure each = α. Then A is one of the angles described by the revolving line OP when, starting from OR, OP stops either in the position OP, or in the position OP,; i. e. any one of the angles Taking the halves of these angles in succession, we have the four dotted lines OP OP OP OP, in the figure, as the lines indicating the various values of the From these it is clear that there are two values of sin A 2' equal in magnitude and opposite in sign. Also, that there (1) When A lies between 180° and 180°, prove that (2) When A lies between 180° and 540°, prove that in terms of cos A, when A lies between 180° (4) Prove that when A lies between (4n+1)π and (4n+3) π, n being a positive integer, cos Α 2 (5) Find sin in terms of cos A, when ▲ lies between 41⁄2o and (4n+2), where n is a positive integer. Thus if we are given the value of sin A, (nothing else being known about the angle A), we have four values for 179. To prove Geometrically that, given the value of sin A (nothing else being known about the angle A), there A Α are four values for sin and four values for cos 11. Let a be the least positive angle which has the given sine, and let ROP1, POL in the figure each = a. |