Plane Trigonometry

Formula Involving Two Angles and Multiple Angles

627 sin(𝐴+𝐵)=sin𝐴cos𝐵+cos𝐴sin𝐵 628 sin(𝐴−𝐵)=sin𝐴cos𝐵−cos𝐴sin𝐵 629 cos(𝐴+𝐵)=cos𝐴cos𝐵−sin𝐴sin𝐵 630 cos(𝐴−𝐵)=cos𝐴cos𝐵+sin𝐴sin𝐵

Proof

By (700) and (701), we have sin𝐶=sin𝐴cos𝐵+cos𝐴sin𝐵 and sin𝐶=sin(𝐴+𝐵)by 622 To obtain sin(𝐴−𝐵) change the sign of 𝐵 in (627), and employ (623), (624), cos(𝐴−𝐵)=sin{(90°−𝐴)−𝐵}by 621 Expand by (628), and use (621), (623), (624). For cos(𝐴−𝐵) change the sign of 𝐵 in (629). 631 tan(𝐴+𝐵)=tan𝐴+tan𝐵1−tan𝐴tan𝐵 632 tan(𝐴−𝐵)=tan𝐴−tan𝐵1+tan𝐴tan𝐵 633 cot(𝐴+𝐵)=cot𝐴cot𝐵−1cot𝐴+cot𝐵 634 cot(𝐴−𝐵)=cot𝐴cot𝐵+1cot𝐵−cot𝐴 Obtained from 627-630 635 sin2𝐴=2sin𝐴cos𝐴627. Put 𝐵=𝐴 636 cos2𝐴=cos²𝐴−sin²𝐴 637 cos2𝐴=2cos²𝐴−1 638 cos2𝐴=1−2sin²𝐴629, 613 639 2cos²𝐴=1+cos2𝐴637 640 2sin²𝐴=1−cos2𝐴638 641 sin𝐴2=1−cos𝐴2640 642 cos𝐴2=1+cos𝐴2 643 tan𝐴2=1−cos𝐴1+cos𝐴=1−cos𝐴sin𝐴=sin𝐴1+cos𝐴652, 642, 613 646 cos𝐴=1−tan²𝐴21+tan²𝐴2; sin𝐴=2tan𝐴21+tan²𝐴2643, 613 648 cos𝐴=11+tan𝐴tan𝐴2 649 sin45°+𝐴2=cos45°−𝐴2=1+sin𝐴2641 650 cos45°+𝐴2=sin45°−𝐴2=1−sin𝐴2642 651 tan45°+𝐴2=1+sin𝐴1−sin𝐴=1+sin𝐴cos𝐴=cos𝐴1−sin𝐴 652 tan2𝐴=2tan𝐴1−tan²𝐴631 Put 𝐵=𝐴 653 cot2𝐴=cot²𝐴−12cot𝐴 654 tan(45°+𝐴)=1+tan𝐴1−tan𝐴 655 tan(45°−𝐴)=1−tan𝐴1+tan𝐴631, 632 656 sin3𝐴=3sin𝐴−4sin³𝐴 657 cos3𝐴=4cos³𝐴−3cos𝐴 658 tan3𝐴=3tan𝐴−tan³𝐴1−3tan²𝐴By putting 𝐵=2𝐴 in 627, 629, and 631 659 sin(𝐴+𝐵)sin(𝐴−𝐵)=sin²𝐴−sin²𝐵  =cos²𝐵−cos²𝐴 660 cos(𝐴+𝐵)cos(𝐴−𝐵)=cos²𝐴−sin²𝐵  =cos²𝐵−sin²𝐴 From 627, ⋯ 661 sin𝐴2+cos𝐴2=±1+sin𝐴Proved by squaring. 662 sin𝐴2−cos𝐴2=±1−sin𝐴 663 sin𝐴2=12{1+sin𝐴−1−sin𝐴} 664 cos𝐴2=12{1+sin𝐴+1−sin𝐴} when 𝐴2 lies between −45° and +45°. 665 In the accompanying diagram the signs exhibited in each quadrant are the signs to be prefixed to the two surds in the value of sin𝐴2 according to the quadrant in which 𝐴2 lies.
For cos𝐴2 change the second sign.

Proof

By examining the changes of sign in (661) and (662) by (607). 666 2sin𝐴cos𝐵=sin(𝐴+𝐵)+sin(𝐴−𝐵) 667 2cos𝐴sin𝐵=sin(𝐴+𝐵)−sin(𝐴−𝐵) 668 2cos𝐴cos𝐵=cos(𝐴+𝐵)+cos(𝐴−𝐵) 669 2sin𝐴sin𝐵=cos(𝐴−𝐵)−cos(𝐴+𝐵) 627-630 670 sin𝐴+sin𝐵=2sin𝐴+𝐵2cos𝐴−𝐵2 671 sin𝐴−sin𝐵=2cos𝐴+𝐵2sin𝐴−𝐵2 672 cos𝐴+cos𝐵=2cos𝐴+𝐵2cos𝐴−𝐵2 673 cos𝐴−cos𝐵=2sin𝐴+𝐵2sin𝐴−𝐵2 Obtained by changing 𝐴 into 𝐴+𝐵2, and 𝐵 into 𝐴−𝐵2, in (666-669).
It is advantageous to commit the foregoing formula to memory, in words, thus: 2sincos=sin sum + sin difference, 2cossin=sin sum − sin difference, 2coscos=cos sum + cos difference, 2sinsin=cos difference − cos sum. sin first + sin second = 2sinhalf sum cos half difference, sin first − sin second = 2coshalf sum sin half difference, cos first + cos second = 2coshalf sum cos half difference, cos second − cos first = 2sinhalf sum sin half difference, 674 sin(𝐴+𝐵+𝐶)=sin𝐴cos𝐵cos𝐶+sin𝐵cos𝐶cos𝐴+sin𝐶cos𝐴cos𝐵−sin𝐴sin𝐵sin𝐶 675 cos(𝐴+𝐵+𝐶)=cos𝐴cos𝐵cos𝐶−cos𝐴sin𝐵sin𝐶−cos𝐵sin𝐶sin𝐴−cos𝐶sin𝐴sin𝐵 676 tan(𝐴+𝐵+𝐶)=tan𝐴+tan𝐵+tan𝐶−tan𝐴tan𝐵tan𝐶1−tan𝐵tan𝐶−tan𝐶tan𝐴−tan𝐴tan𝐵 Proof: Put 𝐵+𝐶 for 𝐵 in 627, 629, and 631. 677 If 𝐴+𝐵+𝐶=180°, sin𝐴+sin𝐵+sin𝐶=4cos𝐴2cos𝐵2cos𝐶2 sin𝐴+sin𝐵−sin𝐶=4sin𝐴2sin𝐵2cos𝐶2 678 cos𝐴+cos𝐵+cos𝐶=4sin𝐴2sin𝐵2sin𝐶2+1 cos𝐴+cos𝐵−cos𝐶=4cos𝐴2cos𝐵2sin𝐶2−1 679 tan𝐴+tan𝐵+tan𝐶=tan𝐴tan𝐵tan𝐶 680 cot𝐴2+cot𝐵2+cot𝐶2=cot𝐴2cot𝐵2cot𝐶2 681 sin2𝐴+sin2𝐵+sin2𝐶=4sin𝐴sin𝐵sin𝐶 682 cos2𝐴+cos2𝐵+cos2𝐶=−4cos𝐴cos𝐵cos𝐶−1 683 General formula, including the foregoing, obtained by applying (666-673).
If 𝐴+𝐵+𝐶=𝜋, and 𝑛 be any integer,
4sin𝑛𝐴2sin𝑛𝐵2sin𝑛𝐶2=sin𝑛𝜋2−𝑛𝐴+sin𝑛𝜋2−𝑛𝐵+sin𝑛𝜋2−𝑛𝐶−sin𝑛𝜋2 684 4cos𝑛𝐴2cos𝑛𝐵2cos𝑛𝐶2=cos𝑛𝜋2−𝑛𝐴+cos𝑛𝜋2−𝑛𝐵+cos𝑛𝜋2−𝑛𝐶+cos𝑛𝜋2 685 If 𝐴+𝐵+𝐶=0, 4sin𝑛𝐴2sin𝑛𝐵2sin𝑛𝐶2=−sin𝑛𝐴−sin𝑛𝐵−sin𝑛𝐶 686 4cos𝑛𝐴2cos𝑛𝐵2cos𝑛𝐶2=cos𝑛𝐴+cos𝑛𝐵+cos𝑛𝐶+1 Rule: If, in formula (683) to (686), two factors on the left be changed by writing sin for cos, or cos for sin, then, on the right side, change the signs of those terms which do not contain the angles of the altered factors. 687 Thus, from (693), we obtain 4sin𝑛𝐴2cos𝑛𝐵2cos𝑛𝐶2=−sin𝑛𝜋2−𝑛𝐴+sin𝑛𝜋2−𝑛𝐵+sin𝑛𝜋2−𝑛𝐶+sin𝑛𝜋2 688 A Formula for the construction of Tables of sines, cosines, ⋯. sin(𝑛+1)𝛼−sin𝑛𝛼=sin𝑛𝛼−sin(𝑛−1)𝛼−𝑘sin𝑛𝛼 where 𝛼=10ʺ, and 𝑘=2(1−cos𝛼)=.0000000023504. 689 Formula for verifying the tables: sin𝐴+sin(72°+𝐴)−sin(72°−𝐴)=sin(36°+𝐴)−sin(36°−𝐴) cos𝐴+cos(72°+𝐴)−cos(72°−𝐴)=cos(36°+𝐴)−cos(36°−𝐴) sin(60°+𝐴)−sin(60°−𝐴)=sin𝐴

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive