Answer to Question #137273 in Trigonometry for riz

Question

1.How to draw/interpret the Napier’s circle?
2.What is the meaning of Napier’s Formula?
3. How to formulate Napier’s Formula?
4.How do you derive formula from the formulated Napier’s Formula?

Answer 1

1.Draw a right triangle on a sphere and label the sides $a, b$ and $c$  where $c$  is the hypotenuse. Let A be the angle opposite side $a$ , B the angle opposite side $b$ , and C the right angle opposite the hypotenuse $c$ .

2.Then Napier’s Formula has two rules: The sine of a part is equal to the product of the tangents of the two adjacent parts. The sine of a part is equal to the product of the cosines of the two opposite parts.

3.In any triangle ABC,

(i) $tan (\frac{B−C} {2} ) = (\frac{b−C} {b+c} ) cot\frac{A} {2}$

(ii) $tan (\frac{C−A} {2} ) = (\frac{c−a} {c+a} ) cot \frac{B} {2}$

(iii) $tan (\frac{A−B} {2} ) = (\frac{a−b} {a+b} ) cot \frac{C} {2}$

4.Considering $c$ , whose non-adjacent parts are 𝑎

and 𝑏.By rule (b), cos c = sin 𝑎sin 𝑏, i.e.,cos c = cos a cos b.This becomes obvious from the cosine rule (1), since γ = 90° and, thus, sin a sin b cos γ = 0. In the same way rule (a) for the adjacent parts α and β of c reads

cos c = cot α cot β.