Find the exact value of arccos(0.33).

Step-by-step solution

1. 1

   Read what the inverse cosine is asking for. $\arccos(0.33)$ is the unique angle $\theta$ in the principal range $[0,\pi]$ (that is $[0^\circ,180^\circ]$) with $\cos\theta = 0.33.$ The restriction to $[0,\pi]$ is what makes the answer a single number rather than an infinite family.

2. 2

   Check whether $0.33$ is a special value. The cosines that have closed forms in elementary trigonometry are the ones built from $0,\pm\tfrac12,\pm\tfrac{\sqrt2}{2},\pm\tfrac{\sqrt3}{2},\pm1$ (and a few nested radicals for multiples of $3^\circ$). Since $0.33$ is none of those, no combination of radicals and $\pi$ will collapse this angle.

3. 3

   Conclude that the exact value is the inverse-cosine expression itself. The honest exact answer is therefore $\theta=\arccos(0.33),$ left in inverse form. Writing something like $\tfrac{\pi}{2}-0.33$ would be wrong: that identity holds for $\arcsin$, not for a stray decimal.

4. 4

   Get the decimal value in radians. Evaluating the inverse cosine numerically, $\arccos(0.33)=1.234492751\ldots \approx 1.234493\ \text{rad}.$ A sanity check: $\arccos(0)=\tfrac{\pi}{2}\approx 1.5708$ and $\arccos(0.5)=\tfrac{\pi}{3}\approx 1.0472$, and $0.33$ lies between $0$ and $0.5$, so the angle must lie between $1.0472$ and $1.5708$ - it does.

5. 5

   Convert to degrees. Multiply by $180/\pi$: $1.234492751\times\frac{180}{\pi}=70.7312245\ldots\approx 70.7312^\circ.$ Verifying in the other direction, $\cos(70.7312^\circ)=0.330000$, which closes the loop.

6. 6

   Watch the rounding. Truncating too early is the common slip here: $70.51^\circ$ corresponds to $\cos\theta=0.3327$, not $0.33$, so quoting $70.51^\circ$ is off by more than two hundredths of a degree in a way that shows up immediately when you take the cosine back.