Find the exact value of each real number y. Do not use a calculator.
y = sin⁻¹ √2/2

검증된 단계별 안내

Recognize that the problem asks for the exact value of \(y = \sin^{-1} \left( \frac{\sqrt{2}}{2} \right)\), which means we need to find the angle \(y\) whose sine is \(\frac{\sqrt{2}}{2}\).

Recall the range of the inverse sine function \(\sin^{-1} x\), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This means the angle \(y\) must lie within this interval.

Identify the common angles where sine values are known exactly. The sine of \(\frac{\pi}{4}\) (or 45 degrees) is \(\frac{\sqrt{2}}{2}\).

Since \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\frac{\pi}{4}\) lies within the range of \(\sin^{-1}\), conclude that \(y = \frac{\pi}{4}\).

Express the final answer as \(y = \frac{\pi}{4}\), which is the exact value of \(\sin^{-1} \left( \frac{\sqrt{2}}{2} \right)\) without using a calculator.

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주요 개념

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

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, sin⁻¹, returns the angle whose sine value is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2]. Understanding this helps find the angle y such that sin(y) = √2/2.

Exact Values of Sine for Special Angles

Certain angles have well-known sine values expressed in exact radicals, such as sin(π/4) = √2/2. Recognizing these special angles allows you to determine the exact angle corresponding to a given sine value without a calculator.

Domain and Range Restrictions of Inverse Trigonometric Functions

Inverse trig functions have restricted ranges to ensure they are functions. For sin⁻¹, the output angle y must lie between -π/2 and π/2. This restriction ensures a unique solution when finding y = sin⁻¹(√2/2).

Find the exact value of each real number y. Do not use a calculator.y = sin⁻¹ √2/2