Question Number 185637 by Mingma last updated on 24/Jan/23
Answered by HeferH last updated on 24/Jan/23
![(a/b) = ((1 + (√5))/2) ((a + b)/(sin (18°+x))) = (a/(sin 18°)) 1 + (b/a) = ((sin (18° + x))/(sin 18°)) 1 + (2/(1 + (√5))) = ((sin (18° + x))/(sin 18°)) sin^(−1) [(1 + (2/(1 + (√5))))sin 18°] − 18° = x x = 12°](Q185644.png)
$$\:\frac{{a}}{{b}}\:=\:\frac{\mathrm{1}\:+\:\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\:\frac{{a}\:+\:{b}}{\mathrm{sin}\:\left(\mathrm{18}°+{x}\right)}\:=\:\frac{{a}}{\mathrm{sin}\:\mathrm{18}°} \\ $$$$\:\mathrm{1}\:+\:\frac{{b}}{{a}}\:=\:\frac{\mathrm{sin}\:\left(\mathrm{18}°\:+\:{x}\right)}{\mathrm{sin}\:\mathrm{18}°}\: \\ $$$$\:\mathrm{1}\:+\:\frac{\mathrm{2}}{\mathrm{1}\:+\:\sqrt{\mathrm{5}}}\:=\:\frac{\mathrm{sin}\:\left(\mathrm{18}°\:+\:{x}\right)}{\mathrm{sin}\:\mathrm{18}°}\: \\ $$$$\mathrm{sin}^{−\mathrm{1}} \:\left[\left(\mathrm{1}\:+\:\frac{\mathrm{2}}{\mathrm{1}\:+\:\sqrt{\mathrm{5}}}\right)\mathrm{sin}\:\mathrm{18}°\right]\:−\:\mathrm{18}°\:=\:{x}\: \\ $$$$\:{x}\:=\:\mathrm{12}°\: \\ $$
Commented by HeferH last updated on 24/Jan/23
