Given sin(theta) = 0.67, find the angle theta in degrees and in radians.

Step-by-step solution

1. 1

   Undo the sine with the inverse sine.

   $\theta=\sin^{-1}(0.67)$

   Arcsine returns the unique angle in $[-90^\circ,90^\circ]$ whose sine is the given value.

2. 2

   Confirm the value is in range. $|0.67|\le1$, so a real angle exists.

3. 3

   Bracket the answer first. $\sin30^\circ=0.5$ and $\sin45^\circ\approx0.7071$. Since $0.5<0.67<0.7071$, the angle must lie between $30^\circ$ and $45^\circ$, closer to $45^\circ$. Anyone who gets $0.73$ has the calculator in radian mode.

4. 4

   Evaluate in degrees.

   $\theta=\sin^{-1}(0.67)=42.0671^\circ\approx42.07^\circ$

   Inside the predicted window and just under $45^\circ$ ✓.

5. 5

   Convert to radians.

   $42.0671\times\frac{\pi}{180}=0.7342\ \text{rad}$

   Just under $\tfrac{\pi}{4}=0.7854$, consistent with an angle just under $45^\circ$.

6. 6

   Note that 0.67 is not a special value, then verify. Unlike $\tfrac12$, $\tfrac{\sqrt2}{2}$ or $\tfrac{\sqrt3}{2}$, the number $0.67$ matches no standard triangle, so there is no exact closed form and a calculator is genuinely required. Checking backwards: $\sin(0.7342088)=0.670000$ ✓.