Find the exact value of arcsin(0.395), or explain why none exists, and give the decimal value in radians and degrees.

Step-by-step solution

1. 1

   State what is being asked. We want the unique $\theta \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ with $\sin\theta = 0.395 = \tfrac{79}{200}$. Since $0 < 0.395 < 1$, the angle is in the first quadrant.

2. 2

   Rule out a closed form. A rational sine gives a closed-form angle only for the handful of values $0,\ \pm\tfrac12,\ \pm 1$ among rationals (Niven's theorem), plus the quadratic surds $\pm\tfrac{\sqrt2}{2}, \pm\tfrac{\sqrt3}{2}$. Because $\tfrac{79}{200}$ is rational but not $0, \pm\tfrac12$ or $\pm1$, the angle is not a rational multiple of $\pi$ and has no elementary exact form. The exact answer is $\arcsin(0.395)$ itself.

3. 3

   Get the radian value from the series. With $u = 0.395$ in $\arcsin u = u + \tfrac{u^3}{6} + \tfrac{3u^5}{40} + \tfrac{15u^7}{336} + \cdots$: $0.395 + 0.010272 + 0.000721 + 0.000067 + 0.000007 = 0.406067$, converging to

   $\arcsin(0.395) \approx 0.4060679\ \text{rad}$

4. 4

   Convert to degrees.

   $0.4060679 \times \frac{180}{\pi} \approx 23.2660^{\circ}$

5. 5

   Check the answer and compare with a neighbour. $\sin(23.2660^{\circ}) = 0.395000$. Comparing with $\arcsin(0.375) \approx 22.0243^{\circ}$: an increase of $0.02$ in the sine moved the angle by about $1.24^{\circ}$, consistent with $\tfrac{d\theta}{d(\sin\theta)} = \tfrac{1}{\cos\theta} \approx 1.088$ radians per unit, i.e. $\approx 62^{\circ}$ per unit sine.