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Geometry Level 2

tan ⁡ ( 6 ∘ ) tan ⁡ ( 5 4 ∘ ) tan ⁡ ( 6 6 ∘ ) = ? \large \tan(6^\circ) \tan(54^\circ) \tan(66^\circ) = \ ?

cot ⁡ ( 1 2 ∘ ) \cot(12^\circ) tan ⁡ ( 2 4 ∘ ) \tan(24^\circ) tan ⁡ ( 1 8 ∘ ) \tan(18^\circ) tan ⁡ ( 1 2 ∘ ) \tan(12^\circ)

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Sarthak Chaudhary by Sarthak Chaudhary , 21, India

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1 solution

Sudeep Salgia
Jul 8, 2015

Using the well known identity, tan ⁡ ( 6 0 ∘ − θ ) tan ⁡ ( θ ) tan ⁡ ( 6 0 ∘ + θ ) = tan ⁡ ( 3 θ ) \displaystyle \tan (60^{\circ } - \theta)\tan (\theta)\tan (60^{\circ } + \theta) = \tan (3 \theta) with θ = 6 ∘ \theta = 6^{\circ } , we get the answer as tan ⁡ 1 8 ∘ \displaystyle \boxed{ \tan 18^{\circ }} .

Note that the θ \theta in the formula has to be taken in degrees ( ∘ ) ( ^{\circ } ) .

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Moderator note:

Yes, Triple Angle Identities solves this problem very quickly.

Bonus question 1 : Find the exact value of tan ⁡ ( 1 8 ∘ ) \tan(18^\circ) .

Bonus question 2 : Find the exact value of tan ⁡ ( 6 ∘ ) , tan ⁡ ( 5 4 ∘ ) , tan ⁡ ( 6 6 ∘ ) \tan(6^\circ),\tan(54^\circ) ,\tan(66^\circ) .

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