Tangents And Tables

Geometry Level 1

tan ⁡ 2 5 ∘ tan ⁡ 3 5 ∘ tan ⁡ 4 5 ∘ tan ⁡ 5 5 ∘ tan ⁡ 6 5 ∘ \tan 25^\circ \qquad \tan 35^\circ \qquad \tan 45^\circ \qquad \tan55^\circ \qquad \tan 65^\circ

The above shows 5 numbers. The reciprocal of each of these numbers is equal to another number in this list, except for one. Which one?

tan ⁡ 2 5 ∘ \tan25^\circ tan ⁡ 3 5 ∘ \tan35^\circ tan ⁡ 4 5 ∘ \tan45^\circ tan ⁡ 5 5 ∘ \tan55^\circ tan ⁡ 6 5 ∘ \tan65^\circ

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Tom Engelsman
Jul 9, 2016

Using the identity tan(x) = cot(90 - x) = 1/tan(90 - x), where x and 90 - x are complementary angles (in degrees), the 45-degree angle will only equal its own reciprocal.

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