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Question Number 159005 by Tawa11 last updated on 11/Nov/21

	Answered by ajfour last updated on 12/Nov/21
	$s=2rsin α p=rsin 2α cos β=((35/2)/r) =(7/8); sin β=((√(15))/8) ((sin (90°−α))/(27))=((sin (3α+β))/p) 20sin 2αcos α=27sin (3α+β) 10sin α+37sin 3α=27sin (3α+β)+27sin 3α =27cos β{3sin α−4sin^3 α +((√(15))/7)(4cos^3 α−3cos α)} say tan α=m 40=27×(7/8){3(1+m^2 )−4m^3 +((√(15))/7)[4−3(1+m^2 )]} ...we need a straight cubic formula...still trying!$
	$s = 2 r \sin α$ $p = r \sin 2 α$ $\cos β = \frac{35 / 2}{r} = \frac{7}{8}; \sin β = \frac{\sqrt{15}}{8}$ $\frac{\sin (90 ° - α)}{27} = \frac{\sin (3 α + β)}{p}$ $20 \sin 2 α \cos α = 27 \sin (3 α + β)$ $10 \sin α + 37 \sin 3 α = 27 \sin (3 α + β) + 27 \sin 3 α$ $= 27 \cos β {3 \sin α - 4 \sin^{3} α$ $+ \frac{\sqrt{15}}{7} (4 \cos^{3} α - 3 \cos α)}$ $s a y \tan α = m$ $40 = 27 \times \frac{7}{8} {3 (1 + m^{2}) - 4 m^{3}$ $+ \frac{\sqrt{15}}{7} [4 - 3 (1 + m^{2})]}$ $. . . w e n e e d a s t r a i g h t c u b i c$ $f o r m u l a . . . s t i l l t r y i n g!$
	Commented by Tawa11 last updated on 12/Nov/21

	$God bless you sir .$
	Answered by mr W last updated on 11/Nov/21

	Commented by mr W last updated on 12/Nov/21

	$s_{1} s_{2} = 27 \times 8$ $s_{1} + s_{2} = s$ $s_{2} + \frac{216}{s_{2}} = s$ $s_{2}^{2} - {s s}_{2} = - 216$ $s_{2} = \frac{s + \sqrt{s^{2} - 864}}{2}$ $s = 2 \times 20 \cos \frac{θ}{2}$ $\Rightarrow \cos \frac{θ}{2} = \frac{s}{40}$ $27^{2} = s^{2} + s_{2}^{2} - 2 {s s}_{2} \cos θ$ $27^{2} = s^{2} + s_{2}^{2} - {s s}_{2} + {s s}_{2} ({2 \cos}^{2} \frac{θ}{2} - 1)$ $27^{2} + 216 = s^{2} + {s s}_{2} ({2 \cos}^{2} \frac{θ}{2} - 1)$ $s^{2} + \frac{s^{2} + s \sqrt{s^{2} - 864}}{2} (\frac{s^{2}}{800} - 1) = 945$ $\Rightarrow s \approx 29.8841$
	Commented by Tawa11 last updated on 12/Nov/21

	$God bless you sir .$
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