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`Plane Trigonometry Use of Subsidiary Angles Sources and References`

# 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−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

ID: 210900009 Last Updated: 9/9/2021 Revision: 0 Ref: References

1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science &amp; Engineering
2. Ayres, F. JR, Moyer, R.E., 1999, Schaum's Outlines: Trigonometry
3. Hopkings, W., 1833, Elements of Trigonometry  Home 5

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