JEE Novice - (14)

Geometry Level 3

tan ⁡ ( 107 n ) ∘ = cos ⁡ 9 6 ∘ + sin ⁡ 9 6 ∘ cos ⁡ 9 6 ∘ − sin ⁡ 9 6 ∘ \Large \tan (107n)^{\circ} = \dfrac{\cos 96^{\circ} + \sin 96^{\circ}}{\cos 96^{\circ} - \sin 96^{\circ}}

Find the smallest positive integer n n that satisfies the above equation.

This question is a part of JEE Novices .


The answer is 3.

Nihar Mahajan by Nihar Mahajan , 20, India

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2 solutions

Chew-Seong Cheong
May 27, 2015

cos ⁡ 9 6 ∘ + sin ⁡ 9 6 ∘ cos ⁡ 9 6 ∘ − sin ⁡ 9 6 ∘ = 1 2 cos ⁡ 9 6 ∘ + 1 2 sin ⁡ 9 6 ∘ 1 2 cos ⁡ 9 6 ∘ − 1 2 sin ⁡ 9 6 ∘ = sin ⁡ 4 5 ∘ cos ⁡ 9 6 ∘ + cos ⁡ 4 5 ∘ sin ⁡ 9 6 ∘ cos ⁡ 4 5 ∘ cos ⁡ 9 6 ∘ − sin ⁡ 4 5 ∘ sin ⁡ 9 6 ∘ = sin ⁡ 14 1 ∘ cos ⁡ 14 1 ∘ = tan ⁡ 14 1 ∘ = tan ⁡ ( 14 1 ∘ + 18 0 ∘ ) = tan ⁡ 32 1 ∘ = tan ⁡ ( 10 7 ∘ × 3 ) ⇒ n = 3 \begin{aligned} \dfrac{\cos{96^\circ}+\sin{96^\circ}}{\cos{96^\circ}-\sin{96^\circ}} & = \frac{\frac{1}{\sqrt{2}} \cos{96^\circ}+\frac{1}{\sqrt{2}} \sin{96^\circ}}{\frac{1}{\sqrt{2}} \cos{96^\circ}-\frac{1}{\sqrt{2}} \sin{96^\circ}} \\ & = \frac{\sin{45^\circ} \cos{96^\circ}+\cos{45^\circ} \sin{96^\circ}}{\cos{45^\circ} \cos{96^\circ}-\sin{45^\circ} \sin{96^\circ}} \\ & = \dfrac {\sin{141^\circ}}{\cos{141^\circ}} = \tan{141^\circ} = \tan{(141^\circ+180^\circ)} \\ & = \tan{321^\circ} = \tan{(107^\circ\times 3)} \\ \Rightarrow n & = \boxed{3} \end{aligned}

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Moderator note:

More generally (in a similar manner as shown above),

tan ⁡ ( 4 5 ∘ + α ) = cos ⁡ α + sin ⁡ α cos ⁡ α − sin ⁡ α . \tan ( 45 ^ \circ + \alpha) = \frac{ \cos \alpha + \sin \alpha } { \cos \alpha - \sin \alpha } .

Since tan ⁡ 32 1 ∘ = tan ⁡ ( − 3 9 ∘ ) \tan{321^\circ} = \tan{(-39^\circ)} , n n can be a negative real number and does not have a defined minimum. It must be specified as an integer.

Chew-Seong Cheong - 6 years ago

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If n is an integer , it does not have a minimum value. As (321-180*107k) can be equal to 107n for every integer k. it must be said that n is a natural number

Srikeshav Kothapalli - 6 years ago

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Yes, I was thinking about it.

Chew-Seong Cheong - 6 years ago
Ravi Dwivedi
Jul 5, 2015

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Moderator note:

Good observation with the tangent sum and product formula.

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