Understanding the arcsine function: Definition, Range, and Calculation

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The arcsine function, denoted as “arcsin(x)” or “sin^(-1)(x)”, is the inverse function of the sine function

The arcsine function, denoted as “arcsin(x)” or “sin^(-1)(x)”, is the inverse function of the sine function. It represents the angle whose sine is a given number “x”.

The arcsine function returns values in the range [-π/2, π/2] (or -90° to 90° in degrees). This is because the sine function repeats its values after every 2π radians (or 360° in degrees), and the arcsine function is the inverse of the sine function in this restricted domain.

To find the arcsine of a number, you can use a scientific calculator with the “arcsin” or “sin^(-1)” function key. Alternatively, you can use the following steps if you do not have a calculator:

1. Start with a number “x” that you want to find the arcsine of.
2. Verify that the number “x” is within the range [-1, 1]. If “x” is outside this range, then arcsine is undefined.
3. Assume that the arcsine of “x” is equal to an angle “y”.
4. Use the definition of the sine function to set up the equation sin(y) = x.
5. Solve this equation for “y” by taking the inverse sine of both sides: y = arcsin(x).
6. Convert the result from radians to degrees if necessary.

For example, let’s find the arcsine of 0.5:
1. We have x = 0.5.
2. Since 0.5 is within the range [-1, 1], it is valid.
3. Let y be the angle whose sine is 0.5.
4. We have sin(y) = 0.5.
5. Taking the inverse sine of both sides gives y = arcsin(0.5).
6. Using a calculator, we find arcsin(0.5) ≈ 30°.

Therefore, the arcsine of 0.5 is approximately 30°.

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