Tan Signs

Geometry Level 2

tan ( 10 0 ) + 4 sin ( 10 0 ) = ? \tan(100^\circ) + 4\sin(100^\circ ) = \ ?

3 \sqrt 3 2 2 3 -\sqrt 3 1 1

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1 solution

Dhirendra Singh
Apr 8, 2015

tan ( 100 ) + 4 × sin ( 100 ) = cot ( 10 ) + 4 × cos ( 10 ) = cos ( 10 ) sin ( 10 ) + 4 × cos ( 10 ) = 4 × cos ( 10 ) × sin ( 10 ) cos ( 10 ) sin ( 10 ) = 2 × sin ( 20 ) cos ( 10 ) sin ( 10 ) = 2 × sin ( 20 ) sin ( 80 ) sin ( 10 ) = sin ( 20 ) + sin ( 20 ) sin ( 80 ) sin ( 10 ) = sin ( 20 ) 2 cos ( 50 ) sin ( 30 ) sin ( 10 ) = sin ( 20 ) cos ( 50 ) sin ( 10 ) = sin ( 20 ) sin ( 40 ) sin ( 10 ) = 2 cos ( 30 ) sin ( 10 ) sin ( 10 ) = 3 \tan (100) + 4\times\sin (100) \\=-\cot(10)+4\times\cos(10) \\=-\dfrac{\cos(10)}{\sin(10)}+4\times \cos(10) \\=\dfrac{4\times\cos(10)\times\sin(10)-\cos(10)}{\sin(10)} \\=\dfrac{2\times\sin(20)-\cos(10)}{\sin(10)} \\=\dfrac{2\times\sin(20)-\sin(80)}{\sin(10)} \\=\dfrac{\sin(20)+\sin(20)-\sin(80)}{\sin(10)} \\=\dfrac{\sin(20)-2\cdot\cos(50)\sin(30)}{\sin(10)} \\=\dfrac{\sin(20)-\cos(50)}{\sin(10)} \\=\dfrac{\sin(20)-\sin(40)}{\sin(10)} \\=\dfrac{-2\cdot\cos(30)\cdot\sin(10)}{\sin(10)} \\=-\sqrt{3}

Moderator note:

Fantastic!

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