Given sin(theta) = 0.66, find the angle theta in degrees and in radians.

Step-by-step solution

1. 1

   Take the inverse sine of both sides.

   $\theta=\sin^{-1}(0.66)$

   The principal value lies in $[-90^\circ,90^\circ]$, and since $0.66>0$ the angle is acute.

2. 2

   Check the domain. $|0.66|\le1$, so the inverse sine is defined.

3. 3

   Estimate before computing. $0.66$ sits between $\sin30^\circ=0.5$ and $\sin45^\circ\approx0.7071$, so expect an angle in the low $40^\circ$s.

4. 4

   Evaluate in degrees.

   $\theta=\sin^{-1}(0.66)=41.2999^\circ\approx41.30^\circ$

5. 5

   Convert to radians.

   $41.2999\times\frac{\pi}{180}=0.7208\ \text{rad}$

6. 6

   Observe the sensitivity, then verify. Comparing with $\sin^{-1}(0.67)=42.07^\circ$: a change of just $0.01$ in the sine shifts the angle by about $0.77^\circ$. That amplification is why inverse-trig inputs should be carried to full precision rather than rounded early. Checking backwards, $\sin(0.7208188)=0.660000$ ✓.