Find the exact value of each real number y. Do not use a calculator.
y = tan⁻¹ (―√3)

검증된 단계별 안내

Recognize that the problem asks for the exact value of \(y = \tan^{-1}(-\sqrt{3})\), which means we need to find an angle \(y\) whose tangent is \(-\sqrt{3}\).

Recall the basic angles where tangent values are known: \(\tan(\frac{\pi}{3}) = \sqrt{3}\) and \(\tan(-\frac{\pi}{3}) = -\sqrt{3}\). These angles are commonly used in trigonometry and correspond to 60° and -60°, respectively.

Since the inverse tangent function \(\tan^{-1}(x)\) returns values in the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (or \((-90^\circ, 90^\circ)\)), the angle \(y\) must lie within this range.

Identify that \(y = -\frac{\pi}{3}\) is the angle in the principal range of \(\tan^{-1}\) such that \(\tan(y) = -\sqrt{3}\).

Therefore, the exact value of \(y\) is \(-\frac{\pi}{3}\), which corresponds to \(-60^\circ\) in degrees.

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주요 개념

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Inverse Tangent Function (arctan)

The inverse tangent function, denoted as tan⁻¹ or arctan, returns the angle whose tangent is a given number. It maps real numbers to angles typically in the range (-π/2, π/2). Understanding this helps find the angle y such that tan(y) equals the given value.

Exact Values of Tangent for Special Angles

Certain angles have well-known exact tangent values, such as π/6, π/4, and π/3. For example, tan(π/3) = √3 and tan(π/6) = 1/√3. Recognizing these values allows one to identify the angle corresponding to a given tangent without a calculator.

Sign and Quadrant Considerations for arctan

Since tan⁻¹ returns angles in (-π/2, π/2), negative tangent values correspond to negative angles in this interval. For tan⁻¹(-√3), the angle is negative and matches the reference angle where tangent is √3, ensuring the correct sign and quadrant are chosen.

Find the exact value of each real number y. Do not use a calculator.y = tan⁻¹ (―√3)