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Calculate sin−1

x= opposite_side/hypotenuse




sin−1(x)

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Inverse Sine sin−1() is used to find the angle for a given ratio of the length of the opposite side to the length of the hypotenuse, in a right angled triangle.

Formula

To find the Inverse Sine of a given ratio: opposite_side/hypotenuse , use the following formula.

θ = sin−1(Length_of_Opposite_side / Length_of_Hypotenuse)

Calculating Sin Inverse

The inverse sine (sin-1 or arcsin) of a value is the angle whose sine is the given value. It is an important function in trigonometry, often used to find angles when the sine value is known.

The inverse sine of a value \( y \) is defined as:

\( \sin^{-1}(y) = \theta \)

where:

Example 1

Let's calculate the inverse sine of 0.5:

The angle whose sine is 0.5 is 30 degrees or \( \frac{\pi}{6} \) radians. So:

\( \sin^{-1}(0.5) = 30^{\circ} = \frac{\pi}{6} \)

Example 2

Consider another example with the value \( \frac{\sqrt{2}}{2} \):

The angle whose sine is \( \frac{\sqrt{2}}{2} \) is 45 degrees or \( \frac{\pi}{4} \) radians. So:

\( \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45^{\circ} = \frac{\pi}{4} \)