Question

find value of tan 7.5

Answer 1

tan 7.5 = 2.7060.....

Answer 2

Solution:-

$\to \rm \: we \: can \: write \: \:$

$\tan7.2 = \tan7 \dfrac{1}{2}$

Let

$\rm \to \: \theta =7 \dfrac{1}{2} \: then, \: 2 \theta = 15 \degree$

Now

$\tan \theta = \dfrac{1 - \cos2 \theta}{ \sin2 \theta }$

$\rm \: [ \because1 - \cos2 \theta = 2 \sin {}^{2} \theta \: and \: \sin2 \theta = 2 \sin\theta \cos\theta]$

$= \dfrac{1 - \cos 15 \degree}{ \sin15 \degree }$

We know that value of

$\to \cos15 \degree = \dfrac{ \sqrt{3} + 1}{2 \sqrt{2} }$

$\to \sin 15 \degree = \dfrac{ \sqrt{3} - 1}{2 \sqrt{2} }$

put the value we get

$= \dfrac{1 - \dfrac{ \sqrt{3} + 1}{2 \sqrt{2} } }{\dfrac{ \sqrt{3} - 1}{2 \sqrt{2} } }$

$\rm = \dfrac{2 \sqrt{2} - \sqrt{3} - 1 }{ \sqrt{3} - 1} = \dfrac{(2 \sqrt{2} - \sqrt{3} - 1)( \sqrt{3} + 1)}{( \sqrt{3} - 1)( \sqrt{3} + 1)}$

$= \dfrac{2 \sqrt{6} - 3 - \sqrt{3 } + 2 \sqrt{2} - \sqrt{3} - 1 }{3 - 1}$

$= \dfrac{2 \sqrt{6} - 2 \sqrt{3} - 4 + 2 \sqrt{2} }{2}$

$= \sqrt{6} - \sqrt{3} - 2 + \sqrt{2}$

$= \sqrt{3} ( \sqrt{2} - 1) - \sqrt{2} ( \sqrt{2} - 1)$

$= (\sqrt{3} - \sqrt{2} )( \sqrt{2} - 1)$

Answer

$= (\sqrt{3} - \sqrt{2} )( \sqrt{2} - 1)$