Find the exact value of each expression.
tan (5π/12)

검증된 단계별 안내

Recognize that the angle \( \frac{5\pi}{12} \) radians is not one of the standard angles with known tangent values, so we need to express it as a sum or difference of angles whose tangent values we know. For example, \( \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} \).

Recall the tangent addition formula: \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]. We will use this formula with \( A = \frac{\pi}{3} \) and \( B = \frac{\pi}{4} \).

Substitute the known tangent values: \( \tan \frac{\pi}{3} = \sqrt{3} \) and \( \tan \frac{\pi}{4} = 1 \) into the formula: \[ \tan \left( \frac{\pi}{3} + \frac{\pi}{4} \right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \times 1} \].

Simplify the numerator and denominator separately to get the expression \( \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \).

To find the exact value, rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator \( (1 + \sqrt{3}) \), then simplify the resulting expression.

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주요 개념

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Angle Sum and Difference Identities

These identities allow the calculation of trigonometric functions for angles expressed as sums or differences of known angles. For tangent, the formula is tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b). This is essential for finding tan(5π/12) by breaking it into angles with known tangent values.

Exact Values of Common Angles

Certain angles like π/4, π/3, and π/6 have well-known exact trigonometric values. Recognizing these angles helps in decomposing complex angles into sums or differences of these standard angles, enabling exact value calculations without a calculator.

Radian Measure

Radian is a unit of angular measure based on the radius of a circle. Understanding how to convert and interpret angles in radians is crucial, as trigonometric functions often use radian inputs, and the problem involves an angle expressed in radians (5π/12).

Find the exact value of each expression.tan (5π/12)