Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.


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tan 𝜋/3

Verified step by step guidance

1. Recall that the angle \( \frac{\pi}{3} \) radians corresponds to 60 degrees, a common special angle in trigonometry.
2. Identify or recall the side lengths of a 30-60-90 right triangle, which are in the ratio 1 : \( \sqrt{3} \) : 2, where the side opposite 60 degrees (\( \frac{\pi}{3} \)) is \( \sqrt{3} \), the side opposite 30 degrees is 1, and the hypotenuse is 2.
3. Use the definition of tangent for an angle in a right triangle: \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \). For \( \theta = \frac{\pi}{3} \), the opposite side is \( \sqrt{3} \) and the adjacent side is 1.
4. Write the expression for \( \tan \frac{\pi}{3} \) as \( \frac{\sqrt{3}}{1} \).
5. Since the denominator is already rational (1), no rationalization is needed. The expression is simplified as is.

Key Concepts

Definition of the Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. For angle θ, tan(θ) = opposite/adjacent. This ratio helps evaluate trigonometric expressions using triangle side lengths.

Special Angles and Their Trigonometric Values

Certain angles like π/3 (60°) have well-known exact trigonometric values. For π/3, tan(π/3) equals √3. Recognizing these special angles allows quick evaluation without needing a calculator or complex calculations.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This process simplifies expressions and is often required for final answers in trigonometry.