Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Sine Function
The inverse sine function, denoted as sin⁻¹ or arcsin, is used to find the angle whose sine is a given number. Its range is restricted to [-π/2, π/2] to ensure that it is a function, meaning each input corresponds to exactly one output. Understanding this function is crucial for solving problems involving angles and their sine values.
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Domain and Range of Sine Function
The sine function has a domain of all real numbers and a range of [-1, 1]. This means that the sine of any angle will always yield a value between -1 and 1. When dealing with the inverse sine function, it is essential to recognize that the input must fall within this range; otherwise, the inverse will not yield a real number.
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Understanding Square Roots
The square root function, denoted as √, returns the non-negative value that, when squared, equals the input number. In the context of the question, √3 is approximately 1.732, which exceeds the range of the sine function. Therefore, since the sine function cannot produce a value greater than 1, the expression sin⁻¹(√3) does not yield a real number.
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