Plane Trigonometry

Properties of The Triangle

700 𝑐=𝑎cos𝐵+𝑏cos𝐴 701 𝑎sin𝐴=𝑏sin𝐵=𝑐sin𝐶 702 𝑎²=𝑏²+𝑐²−2𝑏𝑐cos𝐴

Proof

By Euc. I. 12 and 13, 𝑎²=𝑏²+𝑐²−2𝑐.𝐴𝐷. 703 cos𝐴=𝑏²+𝑐²−𝑎²2𝑏𝑐 704 If 𝑠≡𝑎+𝑏+𝑐2, and △ denote the area 𝐴𝐵𝐶, sin𝐴2=(𝑠−𝑏)(𝑠−𝑐)𝑏𝑐, cos𝐴2=𝑠(𝑠−𝑎)𝑏𝑐641, 642, 703, 9, 10, 1 705 tan𝐴2=(𝑠−𝑏)(𝑠−𝑐)𝑠(𝑠−𝑎) 706 sin𝐴=2𝑏𝑐𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐) 707 △=𝑏𝑐2sin𝐴=𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)707, 706 708 △=142𝑏²𝑐²+𝑐²𝑎²+𝑎²𝑏²−𝑎⁴−𝑏⁴−𝑐⁴

Sources and References

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