Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ 0

검증된 단계별 안내

Understand that the problem asks for the values of \(y\) such that \(\sin y = 0\), where \(y = \sin^{-1} 0\) represents the inverse sine (arcsine) function.

Recall the definition of the inverse sine function: \(y = \sin^{-1} x\) means \(\sin y = x\) and \(y\) lies within the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Set up the equation based on the problem: \(\sin y = 0\) with \(y \in [-\frac{\pi}{2}, \frac{\pi}{2}]\).

Identify all angles \(y\) in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) where \(\sin y = 0\). These are the values where the sine function crosses zero within the principal range.

Conclude that the exact value(s) of \(y\) satisfying the equation are those angles found in the previous step, which represent the output of \(\sin^{-1} 0\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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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, denoted as sin⁻¹ or arcsin, returns the angle whose sine 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 function helps find angles corresponding to specific sine values.

Sine Function Values

The sine function relates an angle to the ratio of the opposite side over the hypotenuse in a right triangle. Key values include sin(0) = 0, sin(π/2) = 1, and sin(-π/2) = -1. Recognizing these standard values aids in identifying angles for given sine outputs.

Domain and Range Restrictions of Inverse Trigonometric Functions

Inverse trigonometric functions have restricted ranges to ensure they are functions. For arcsin, the output is limited to angles between -π/2 and π/2. This restriction means that for sin⁻¹(0), the principal value is 0, even though sine of other angles like π also equals zero.

Find the exact value of each real number y if it exists. Do not use a calculator.y = sin⁻¹ 0