Evaluate arctan(15/66.67), giving the answer in both degrees and radians.

Step-by-step solution

1. 1

   Understand what the inverse tangent returns. $\arctan t$ is the angle $\theta$ with $\tan\theta=t$ and $-90^{\circ}<\theta<90^{\circ}$. Unlike $\arcsin$ and $\arccos$, its domain is all real numbers, so no range check on the input is needed. In applied work this shape — $\arctan\frac{\text{opposite}}{\text{adjacent}}$ — is how a right triangle's acute angle is recovered from its two legs.

2. 2

   Reduce the ratio to a single decimal. Do the division before the trig function:

   $\frac{15}{66.67}=0.2249887\ldots\approx0.2250$

   so the task becomes $\arctan(0.2250)$.

3. 3

   Predict the size of the answer. Since $\tan0^{\circ}=0$ and $\tan 45^{\circ}=1$, a tangent of only $0.225$ must give an angle well under $45^{\circ}$ — and because $\tan\theta\approx\theta$ in radians for small angles, expect roughly $0.225$ rad $\approx13^{\circ}$. This estimate is exactly what catches a calculator left in the wrong angle mode.

4. 4

   Evaluate in radians.

   $\arctan(0.2249887)=0.2213037\ \text{rad}\approx0.2213\ \text{rad}$

   As predicted, slightly below the input $0.225$, since $\tan\theta>\theta$ for small positive $\theta$.

5. 5

   Convert to degrees. Multiply by $\dfrac{180}{\pi}$:

   $0.2213037\times\frac{180}{\pi}=12.6798^{\circ}\approx12.68^{\circ}$

6. 6

   Verify by going backwards. $\tan(12.6798^{\circ})=0.224989$, which matches $\tfrac{15}{66.67}=0.224989$ to six decimals ✓.