sin-50-sin-40-without-tables-or-calculators-

Question Number 73787 by Rio Michael last updated on 15/Nov/19

s i n 50 + s i n 40 = ? w i t h o u t t a b l e s o r c a l c u l a t o r s

Commented by mind is power last updated on 15/Nov/19

i see just cardan Methode

\sin (50) = \sin (90 - 40) = \cos (40)

\Rightarrow \sin (50) + s i n (40) = c o s (40) + s i n (40) = \sqrt{2} c o s (5)

w e f i n d c o s (10)

b y c o s (3.10) = \frac{\sqrt{3}}{2}

c o s (3 x) = 4 {c o s}^{3} (x) - 3 c o s (x) \Rightarrow \cos (10) Root of

p (z) = z^{3} - \frac{3}{4} z - \frac{\sqrt{3}}{8}

ther use cardon

X^{3} + pX + q = 0

particular solution

Δ = q^{2} + \frac{{4 p}^{3}}{27}

X_{1} = \sqrt[3]{\frac{1}{2} (- q + \sqrt{Δ})} + \sqrt[3]{\frac{1}{2} (- q - \sqrt{Δ})}

we want reel root

ther get \cos (10), \cos (5) = \frac{\sqrt{1 + \cos (10)}}{\sqrt{2}}

Commented by Rio Michael last updated on 15/Nov/19

t h a n k s