Find the Exact Value tan(420)

Problem

tan(420^∘)

Solution

1. Find the coterminal angle by subtracting 360^∘ from the given angle to find an equivalent angle within the first rotation.

θ=420^∘−360^∘

θ=60^∘

2. Identify the trigonometric identity for the tangent of a coterminal angle, which states that tan(θ+360^∘⋅n)=tan(θ)

tan(420^∘)=tan(60^∘)

3. Determine the exact value of tan(60^∘) using the properties of a 30^∘−60^∘−90^∘ reference triangle or the unit circle.

tan(60^∘)=√(,3)

Final Answer

tan(420^∘)=√(,3)

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