Beyond triangles

Geometry Level 2

tan ⁡ θ = tan ⁡ 1 5 ∘ tan ⁡ 2 5 ∘ tan ⁡ 3 5 ∘ \large \tan\theta = \tan 15^\circ\tan 25^\circ\tan 35^\circ

If θ \theta is a positive acute angle satisfying the equation above, find the measure of θ \theta in degrees.


The answer is 5.

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Danish Ahmed by Danish Ahmed , 24, India

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1 solution

Danish Ahmed
May 12, 2015

tan ⁡ 1 5 ∘ tan ⁡ 2 5 ∘ tan ⁡ 3 5 ∘ \tan15^{\circ}\tan 25^{\circ} \tan 35^{\circ}

= tan ⁡ 1 5 ∘ ⋅ tan ⁡ 2 5 ∘ tan ⁡ 3 5 ∘ tan ⁡ 8 5 ∘ tan ⁡ 8 5 ∘ =\tan 15^{\circ} \cdot \dfrac{\tan 25^{\circ} \tan 35^{\circ} \tan 85 ^{\circ}}{\tan 85^{\circ}}

= tan ⁡ 1 5 ∘ ⋅ tan ⁡ 7 5 ∘ tan ⁡ 8 5 ∘ =\tan 15^{\circ} \cdot \dfrac{\tan 75^{\circ}}{\tan 85^{\circ}}

= tan ⁡ 1 5 ∘ tan ⁡ 4 5 ∘ tan ⁡ 7 5 ∘ tan ⁡ 4 5 ∘ tan ⁡ 8 5 ∘ =\dfrac{\tan 15^{\circ} \tan 45^{\circ} \tan 75^{\circ}}{\tan 45^{\circ} \tan 85^{\circ}}

= tan ⁡ 4 5 ∘ tan ⁡ 4 5 ∘ tan ⁡ 8 5 ∘ = 1 tan ⁡ 8 5 ∘ = tan ⁡ 5 ∘ ⟹ θ = 5 ∘ =\dfrac{\tan 45^{\circ}}{\tan 45^{\circ} \tan 85^{\circ}}=\dfrac{1}{\tan 85^{\circ}}=\tan 5^{\circ}\implies \theta = 5^{\circ}

Formula used is :

tan ⁡ x tan ⁡ ( 6 0 ∘ − x ) tan ⁡ ( 6 0 ∘ + x ) = tan ⁡ 3 x \tan x\tan(60^{\circ}-x)\tan(60^{\circ}+x)=\tan 3x

11 Helpful 1 Interesting 0 Brilliant 0 Confused

Moderator note:

You could simplify your working by a bit. Hint: 2 5 ∘ + 3 5 ∘ = 6 0 ∘ 25^\circ + 35^\circ = 60^\circ .

Challenge Master:

With tan ⁡ A tan ⁡ ( 60 − A ) tan ⁡ ( 60 + A ) = tan ⁡ ( 3 A ) \tan A \tan(60 - A) \tan(60 + A) = \tan(3A) , set A = 2 5 ∘ A = 25^\circ .

tan ⁡ 2 5 ∘ × tan ⁡ 3 5 ∘ × tan ⁡ 8 5 ∘ = tan ⁡ 7 5 ∘ tan ⁡ 2 5 ∘ × tan ⁡ 3 5 ∘ × cot ⁡ 5 ∘ = cot ⁡ 1 5 ∘ tan ⁡ 2 5 ∘ × tan ⁡ 3 5 ∘ × tan ⁡ 1 5 ∘ = tan ⁡ 5 ∘ \begin{aligned} \tan 25^\circ \times \tan 35^\circ \times \tan 85^\circ &=& \tan 75^\circ \\ \tan 25^\circ \times \tan 35^\circ \times \cot 5^\circ &=& \cot 15^\circ \\ \tan 25^\circ \times \tan 35^\circ \times\tan 15^\circ &=& \tan 5^\circ \end{aligned}

Pi Han Goh - 6 years, 1 month ago

how come we get tan 15*tan 75=tan15 * tan45 * tan 75??? plzz.. I need to know it..!!

Vanshika Bisht - 6 years ago

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What does tan 45 equal? I believe you can multiply by 1 all you like, no matter what form it comes in.

Louis W - 6 years ago

Where did you get that formula? I see how it is derived, but I've never seen it before, it would have been nice to know.

Louis W - 6 years ago

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