Question 1

What is arcsin (inverse sine)?

Accepted Answer

Arcsin, also written as sin⁻¹ or arcsine, is the inverse function of sine. It takes a value between -1 and 1 and returns the angle (in radians or degrees) whose sine is that value. The principal value of arcsin is always between -π/2 and π/2 (or -90° and 90°).

Question 2

How do you calculate arcsin?

Accepted Answer

To calculate arcsin: 1) Ensure your input value is between -1 and 1 (the range of sine). 2) Use the arcsin function (often labeled sin⁻¹ on calculators). 3) The result will be an angle in the range [-π/2, π/2] or [-90°, 90°]. 4) You can convert between radians and degrees using the conversion: degrees = radians × (180/π).

Question 3

What is the domain and range of arcsin?

Accepted Answer

The domain of arcsin is [-1, 1], meaning it only accepts input values between -1 and 1 (inclusive). The range is [-π/2, π/2] in radians or [-90°, 90°] in degrees. This restricted range is called the principal value and ensures arcsin is a proper function (one output for each input).

Question 4

What is arcsin(0), arcsin(1), and arcsin(-1)?

Accepted Answer

Common arcsin values: arcsin(0) = 0 (the angle whose sine is 0 is 0 radians or 0°). arcsin(1) = π/2 ≈ 1.5708 radians or 90° (the maximum value). arcsin(-1) = -π/2 ≈ -1.5708 radians or -90° (the minimum value). These are important reference points.

Question 5

How to calculate arcsin in radians vs degrees?

Accepted Answer

Arcsin can be expressed in either radians or degrees. To convert: From radians to degrees: multiply by 180/π. From degrees to radians: multiply by π/180. Most calculators have a mode setting to choose the output unit. The mathematical definition uses radians, but degrees are often more intuitive.

Question 6

What is the difference between sin and arcsin?

Accepted Answer

Sin (sine) takes an angle and returns a ratio between -1 and 1. Arcsin (inverse sine) does the opposite: it takes a ratio between -1 and 1 and returns an angle. They are inverse functions: sin(arcsin(x)) = x for x in [-1,1], and arcsin(sin(x)) = x only when x is in [-π/2, π/2].

Question 7

Why does arcsin only work for values between -1 and 1?

Accepted Answer

The sine function only produces values between -1 and 1 for any angle input. Since arcsin is the inverse of sine, it can only accept values that sine could have produced, which is the range [-1, 1]. Any value outside this range has no corresponding angle, making arcsin undefined.

Question 8

How to find all solutions using arcsin?

Accepted Answer

Arcsin gives the principal value (one solution). To find all solutions: For arcsin(x) = θ, all solutions are θ + 2πn and (π - θ) + 2πn, where n is any integer. In degrees: θ + 360°n and (180° - θ) + 360°n. This accounts for the periodic nature of sine and the symmetry of the sine curve.

Question 9

What are common arcsin values to memorize?

Accepted Answer

Key arcsin values: arcsin(0) = 0°, arcsin(1/2) = 30°, arcsin(√2/2) = 45°, arcsin(√3/2) = 60°, arcsin(1) = 90°. Negative values: arcsin(-1/2) = -30°, arcsin(-√2/2) = -45°, arcsin(-√3/2) = -60°, arcsin(-1) = -90°. These correspond to special angles on the unit circle.

Question 10

How is arcsin used in real-world applications?

Accepted Answer

Arcsin is used in physics (calculating angles in projectile motion, optics for refraction angles), engineering (structural analysis, wave mechanics), navigation (determining elevation angles), computer graphics (3D rotations), and astronomy (calculating celestial angles). It's essential whenever you need to find an angle from a sine ratio.

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