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Geometry Level 2

Given that the triangle which has sides in the ratio 3 : 4 : 5 3:4:5 , has 2 angles approximately equal to 3 7 ∘ 37^\circ and 5 3 ∘ 53^\circ , which of the following values can be found by using simple trigonometric identities?

sin ⁡ 1 7 ∘ \sin 17^\circ sin ⁡ 1 3 ∘ \sin 13^\circ sin ⁡ 1 4 ∘ \sin 14^\circ sin ⁡ 1 6 ∘ \sin 16^\circ

Aditya Raut by Aditya Raut , 23, India

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1 solution

Omkar Kulkarni
Jan 7, 2015

sin ⁡ ( 1 6 ∘ ) \sin (16^{\circ})

= sin ⁡ ( 5 3 ∘ − 3 7 ∘ ) = \sin (53^{\circ} - 37^{\circ})

= sin ⁡ ( 5 3 ∘ ) cos ⁡ ( 3 7 ∘ ) − cos ⁡ ( 5 3 ∘ ) sin ⁡ ( 3 7 ∘ ) = \sin(53^{\circ})\cos(37^{\circ}) - \cos(53^{\circ})\sin(37^{\circ})

= ( 4 5 ) ( 4 5 ) − ( 3 5 ) ( 3 5 ) = (\frac {4}{5})(\frac {4}{5}) - (\frac {3}{5})(\frac {3}{5})

= 16 25 − 9 25 = \frac {16}{25} - \frac {9}{25}

= 7 25 = \frac {7}{25}

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