Plane Trigonometry

Use of Subsidiary Angles

749 To adapt 𝑎±𝑏 to logarithmic computation. Take 𝜃=tan⁻¹𝑏𝑎 then 𝑎+𝑏=𝑎sec²𝜃 750 For 𝑎−𝑏=𝑎2cos(𝜃+45°)cos𝜃 751 To adapt 𝑎cos𝐶±𝑏sin𝐶 to logarithmic computation. Take 𝜃=tan⁻¹𝑎𝑏, then 𝑎cos𝐶±𝑏sin𝐶=(𝑎²+𝑏²)sin(𝜃±𝐶)By 617 For similar instances of the use of a subsidiary angle, see (726) to (730). 752 To solve a quadratic equation by employing a subsidiary angle. If 𝑥²−2𝑝𝑥+𝑞=0 be the equation, 𝑥=𝑝1±1−𝑞𝑝²By 45 Case I. If 𝑞 be <𝑝², put 𝑞𝑝²=sin²𝜃; then 𝑥=2𝑝cos²𝜃2, and 2𝑝sin²𝜃2By 639, 640 Case II. If 𝑞 be >𝑝², put 𝑞𝑝²=sec²𝜃; then 𝑥=𝑝(1±𝑖tan𝜃), imaginary roots614 Case III. If 𝑞 be negative, put 𝑞𝑝²=tan²𝜃; then 𝑥=𝑞cot𝜃2, and 𝑥=−𝑞cot𝜃2By 644, 645

Sources and References

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