Find the exact value of each real number y if it exists. Do not use a calculator.
y = tan⁻¹ 1

검증된 단계별 안내

Recognize that the expression \(y = \tan^{-1} 1\) asks for the angle \(y\) whose tangent value is 1.

Recall the definition of the inverse tangent function: \(\tan^{-1} x\) gives the angle \(\theta\) such that \(\tan \theta = x\) and \(\theta\) lies within the principal range \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).

Identify the angle within the principal range where the tangent is 1. From the unit circle or common trigonometric values, \(\tan \frac{\pi}{4} = 1\).

Conclude that the exact value of \(y\) is the angle \(\frac{\pi}{4}\), since it satisfies \(\tan y = 1\) and lies in the principal range of \(\tan^{-1}\).

Therefore, the solution is \(y = \frac{\pi}{4}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 function helps find the angle y such that tan(y) equals the given value.

Tangent Values of Special Angles

Certain angles have well-known tangent values, such as tan(π/4) = 1. Recognizing these special angles allows you to find exact values without a calculator by matching the given tangent value to a standard angle.

Range and Principal Values of Inverse Trigonometric Functions

Inverse trig functions have restricted output ranges to ensure they are functions. For arctan, the principal value lies between -π/2 and π/2. This restriction ensures a unique solution for y = tan⁻¹(1), which is important for determining the exact angle.

Find the exact value of each real number y if it exists. Do not use a calculator.y = tan⁻¹ 1