Find the exact value of arcsin(0.375), or explain why none exists, and give the decimal value in radians and degrees.

Step-by-step solution

1. 1

   Write the decimal as a fraction. $0.375 = \tfrac{3}{8}$, so the problem is to find $\theta$ with

   $\sin\theta = \frac{3}{8}, \qquad \theta \in \left[-\tfrac{\pi}{2},\ \tfrac{\pi}{2}\right]$

   The restricted range is what makes $\arcsin$ a function: of the infinitely many angles with this sine, exactly one lies in that interval.

2. 2

   Check it against the list of constructible sines. The angles with a closed-form sine that school work uses are $0,\ \tfrac{\pi}{6},\ \tfrac{\pi}{4},\ \tfrac{\pi}{3},\ \tfrac{\pi}{2}$, whose sines are $0,\ \tfrac12,\ \tfrac{\sqrt2}{2},\ \tfrac{\sqrt3}{2},\ 1$. The value $\tfrac38 = 0.375$ is not one of them — it sits between $0$ and $\tfrac12$, closer to $\tfrac12$.

3. 3

   Conclude what "exact value" means here. No expression in radicals of rationals equals this angle, so the exact value is simply the symbol itself:

   $\arcsin\!\left(\frac{3}{8}\right)$

   This is not a failure of technique — it is the same sense in which $\ln 2$ or $\sqrt2$ is already an exact answer.

4. 4

   Compute the decimal value. Using the Maclaurin series $\arcsin u = u + \tfrac{u^3}{6} + \tfrac{3u^5}{40} + \tfrac{15u^7}{336} + \cdots$ with $u = 0.375$: $0.375 + 0.008789 + 0.000556 + 0.000047 + 0.000004 = 0.384396$, and further terms push it to

   $\arcsin(0.375) \approx 0.3843968\ \text{rad}$

5. 5

   Convert to degrees and check. Multiplying by $\tfrac{180}{\pi}$:

   $0.3843968 \times 57.295780 \approx 22.0243^{\circ}$

   Sanity check: $\sin(22.0243^{\circ}) = 0.375000$, and $22.0243^{\circ}$ is sensibly a little below $30^{\circ}$, where the sine reaches $0.5$.