In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ (-0.32)

검증된 단계별 안내

Identify the function involved: here, \( \sin^{-1} \) represents the inverse sine function, also known as arcsine, which gives the angle whose sine is the given value.

Recall the domain and range of the inverse sine function: the input value must be between -1 and 1, and the output angle will be in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) or equivalently \( [-90^\circ, 90^\circ] \). Since -0.32 is within the domain, the calculation is valid.

Use a calculator set to the correct mode (degrees or radians depending on the problem context) to find \( \sin^{-1}(-0.32) \).

Input the value -0.32 into the inverse sine function on the calculator to get the angle whose sine is -0.32.

Round the resulting angle to two decimal places as required.

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주요 개념

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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 typically in the range of -90° to 90° (or -π/2 to π/2 radians). Understanding this helps in finding the angle corresponding to a sine value.

Domain and Range of the Inverse Sine Function

The domain of sin⁻¹ is limited to values between -1 and 1 because sine values cannot exceed this range. The range of sin⁻¹ is the set of possible output angles, usually from -90° to 90°. Recognizing these limits ensures the input is valid and the output angle is interpreted correctly.

Using a Calculator to Evaluate Inverse Trigonometric Functions

Calculators have specific modes (degree or radian) that affect the output of inverse trig functions. To find sin⁻¹(-0.32), input the value correctly and ensure the calculator is set to the desired angle unit. Rounding the result to two decimal places provides a precise and usable answer.