How do you evaluate $csc 75$ $csc 75$ ?

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Answer 1 · 2016-07-22T03:56:05

$csc 75 = (\sqrt{6} - \sqrt{2}) ≅ 1.0353$ $csc75=(sqrt6-sqrt2)~=1.0353$ ..

To find $csc 75$ $csc75$ , we first find $sin 75 : -$ $sin75 :-$

$sin 75 = sin (45 + 30) = sin 45 cos 30 + cos 45 sin 30$ $sin75=sin(45+30)=sin45cos30+cos45sin30$

$= \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2}$ $=1/sqrt2*sqrt3/2+1/sqrt2*1/2$

$= \frac{\sqrt{3} + 1}{2 \sqrt{2}}$ $=(sqrt3+1)/(2sqrt2)$

$= \frac{\sqrt{6} + \sqrt{2}}{4}$ $=(sqrt6+sqrt2)/4$ .

Hence, $csc 75 = \frac{1}{sin 75}$ $csc75=1/sin75$

$= \frac{4}{\sqrt{6} + \sqrt{2}}$ $=4/(sqrt6+sqrt2)$

$\frac{4 (\sqrt{6} - \sqrt{2})}{(\sqrt{6} + \sqrt{2}) (\sqrt{6} - \sqrt{2})}$ ${4(sqrt6-sqrt2)}/{(sqrt6+sqrt2)(sqrt6-sqrt2)}$

$= (\sqrt{6} - \sqrt{2})$ $=(sqrt6-sqrt2)$ .

Taking, $\sqrt{6} ≅ 2.4495, and, \sqrt{2} ≅ 1.4142$ $sqrt6~=2.4495, and, sqrt2~=1.4142$ , we get,

$csc 75 ≅ 2.4495 - 1.4142 ≅ 1.0353$ $csc75~=2.4495-1.4142~=1.0353$ .

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