Now observing that when the + sign precedes a (i.e. in the first and last of these expressions) the multiple of which is prefixed is even, and either positive or negative, and when the sign precedes a, the multiple of π is odd, and either positive or negative; it is evident that all these angles are represented by the general expression where n stands for any integer, either even or odd, positive or negative. Therefore sin {n+(-1)" a}= sin a. 8. To find a general expression for all angles which have the same given value for their cosine. Let AOP be the least primary angle which has its cosine equal to the given value, and let a be its circular measure. Take AOP' equal to AOP in magnitude. Then since the sign of the cosine is the same for positions of the revolving line in the first and fourth quadrants, the cosine of all angles N corresponding to the positions OP, OP' of the revolving line will be the same and equal to the given value. These angles are evidently primary angles, a, 2π tiple of 2π, α, and these increased by any mul secondary angles, -α, —(2π-a) and these increased by any or 2mπ+a, 2(m+1)π−a, −2mπ − α, − 2(m+1)π+ α, 9. To find a general expression for all angles which have the same given value for their tangent. Denoting the circular measure of PON by a, as before, the two positions of the revolving line for which the tangent is the same as that of a are OP and OP'. The N' primary and secondary angles corresponding to these two positions are represented by and these increased by any multiples of 27, which give + α, N P — 2Mπ — (π — α), — 2Mπ— (2π—a), or, 2mπ+ α, (2m+1)π+ α, −(2m+1)π+ α, −2(m+1)π+ α. Observing now that the sign prefixed to a is always positive while the mutiple of is either even or odd, positive or negative, we see that all these angles are included in the one general formula where n is any integer, odd or even, positive or negative, so that tan a= =tan (nπ + α). 10. It follows evidently from these results that cosec {nπ+(-1)"a} = cosec a, sec (2nπa) = sec a, cot (n+a)=cot a. The results of the three preceding articles must be committed to memory, as they are of continual recurrence. It may be observed that it is not necessary in these expressions that a should be absolutely the least primary angle which has the given value for the trigonometrical functions. They are equally true if a be any angle which has the given value for its sine, cosine, or tangent, as the case may be. CHAPTER IV. TRIGONOMETRICAL FUNCTIONS OF THE SUM AND DIFFERENCE OF TWO ANGLES AND OF THE MULTIPLES AND SUBMULTIPLES OF ANGLES. 1. To find sin (A+B) in terms of the sines and cosines of A and B. to PM, AB respectively. Then the angle RPQ = the angle RQA, for each is the complement of PQR, and therefore the angle RPQ is equal to A. 2. Cos (A+B) in terms of sines and cosines of A and B. Employing the same fig. as in Art. 1, we have AM AN-MN AN QR cos (A+B)= = AP AP AP AP AN AQ QR QP AQAP QP'AP =cos A cos B-sin A sin B. 3. Tan (A+B) in terms of tangents of A and B. PM RM+PR QN+ PR tan (A+B)=AMAN-MN ̄ AN— QR® Dividing the numerator and denominator of this fraction by AN, QN PR + AN AN tan (A+B)= =tan A, and by similar triangles PQR, QAN, 4. Sin (A-B) and cos (A–B) in terms of sines an cosines of A and B. Let BAC be represented by A and CAD by B. Then BAD-A-B. Take any point Pin AD and from P draw PQ at A N M right angles to AC and PM at right angles to AB, and draw QN, PR respectively parallel to PM, AB. Then the angle PQR, being the complement of AQN, QN-QR_QN AQ QR QP =sin A cos B-cos A sin B, =cos A cos B+ sin A sin B. 5. Tan (A-B) in terms of tan A and tan B. AQ AP PQ'AP 6. The expressions for tan (A+B) and tan (A-B) are also easily deduced from those for the sines and cosines of A+B and A-B. AN tan A-tan B or, dividing the numerator and denominator of this frac |