sin-1 (0.6)

sin-1 (0.6)

The correct answer and explanation is:

The value of sin⁡−1(0.6)\sin^{-1}(0.6), also called the inverse sine or arcsine of 0.6, is approximately: sin⁡−1(0.6)≈0.6435 radians\sin^{-1}(0.6) \approx 0.6435 \text{ radians}

In degrees, this is about: 0.6435×180π≈36.87∘0.6435 \times \frac{180}{\pi} \approx 36.87^\circ

Explanation

The inverse sine function, denoted sin⁡−1\sin^{-1} or arcsin⁡\arcsin, is used to find the angle whose sine value equals the given number. The sine function itself gives the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle.

The input to sin⁡−1(x)\sin^{-1}(x) must be within the domain [−1,1][-1, 1], since sine values cannot be less than -1 or greater than 1. Here, 0.6 fits within that range.

When calculating sin⁡−1(0.6)\sin^{-1}(0.6), the output is the angle θ\theta such that sin⁡(θ)=0.6\sin(\theta) = 0.6. The principal range of sin⁡−1\sin^{-1} is from −π2-\frac{\pi}{2} to π2\frac{\pi}{2} radians (or -90° to 90°), which means the inverse sine returns angles in this interval.

Using a calculator or trigonometric tables, the angle whose sine is 0.6 is approximately 0.6435 radians. This corresponds roughly to 36.87 degrees.

This result is important in trigonometry and applied mathematics where angles are calculated from known sine values, for example in physics for projectile motion angles or engineering for forces at certain angles.

Inverse sine is a fundamental trigonometric inverse function and is widely used in geometry, physics, engineering, and computer graphics whenever one needs to determine an angle from a known sine ratio.

To summarize:

• Input: 0.6 (within valid domain)
• Output: angle ≈ 0.6435 radians or ≈ 36.87 degrees
• Meaning: angle whose sine is 0.6

This allows calculation of angles when the ratio of opposite side to hypotenuse is known.