Find the value of arctan(3716/3936), giving the answer in both radians and degrees.

Step-by-step solution

1. 1

   Reduce the fraction before anything else. Both numbers are divisible by $4$:

   $\gcd(3716,3936)=4\ \Longrightarrow\ \frac{3716}{3936}=\frac{929}{984}$

   Reducing first is worth doing because it makes the next step - checking for a special angle - possible to do by eye.

2. 2

   Test whether the input is a special tangent value. The exact-value tangents are $0$, $\tfrac{\sqrt3}{3}=0.5774$, $1$ and $\sqrt3=1.7321$. Here

   $\frac{929}{984}=0.944106$

   which is none of them, and $929$ is prime so no hidden surd is lurking. There is no exact closed form; a calculator is genuinely required.

3. 3

   Bracket the answer before computing. Since $0.944106<1$ and tangent increases on $(0,\tfrac{\pi}{2})$, the angle must be less than $\arctan 1=45^{\circ}$ - but only slightly, since the ratio is close to $1$. Expect something in the low $40$s of degrees.

4. 4

   Evaluate in radians.

   $\arctan(0.944106)=0.7566554\ \text{rad}$

5. 5

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

   $0.7566554\times\frac{180}{\pi}=43.3532^{\circ}$

   This lands just under $45^{\circ}$, exactly as predicted in step 3 ✓.

6. 6

   Verify by taking the tangent back. $\tan(0.7566554)=0.944106$, which reproduces $\tfrac{929}{984}$ to six decimal places ✓. As a further check, $0.7566554$ rad is a little below $\tfrac{\pi}{4}=0.785398$ rad, consistent with the angle being below $45^{\circ}$.