The problem asks to evaluate $\tan(\frac{\pi}{3})$.

TrigonometryTrigonometryTangentUnit CircleRadiansSpecial Angles
2025/3/12

1. Problem Description

The problem asks to evaluate tan(π3)\tan(\frac{\pi}{3}).

2. Solution Steps

We need to find the value of the tangent function at π3\frac{\pi}{3} radians.
Recall that tan(θ)=sin(θ)cos(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.
First, we find sin(π3)\sin(\frac{\pi}{3}) and cos(π3)\cos(\frac{\pi}{3}).
We know that π3\frac{\pi}{3} radians is equal to 6060^\circ.
sin(π3)=sin(60)=32\sin(\frac{\pi}{3}) = \sin(60^\circ) = \frac{\sqrt{3}}{2}
cos(π3)=cos(60)=12\cos(\frac{\pi}{3}) = \cos(60^\circ) = \frac{1}{2}
Therefore,
tan(π3)=sin(π3)cos(π3)=3212=3221=3\tan(\frac{\pi}{3}) = \frac{\sin(\frac{\pi}{3})}{\cos(\frac{\pi}{3})} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \cdot \frac{2}{1} = \sqrt{3}.

3. Final Answer

3\sqrt{3}

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