Evaluate cos(sin(3)), with the argument in radians.

Step-by-step solution

1. 1

   Work from the inside out. This is a composition, not a product: $\cos(\sin 3)$ means take $\sin 3$ first, then feed that number into $\cos$. It is not $\cos3\times\sin3$.

2. 2

   Settle the angle mode before computing. With no degree symbol, $3$ means $3$ radians. Since $\pi\approx3.14159$, an angle of $3$ radians is just short of a half turn — near $171.9^\circ$, in the second quadrant, where sine is small and positive.

3. 3

   Evaluate the inner function.

   $\sin3=0.1411200$

   This is small precisely because $3$ is close to $\pi$, where sine vanishes; indeed $\sin3\approx\sin(\pi-3)=\sin(0.14159)\approx0.1412$ ✓.

4. 4

   Evaluate the outer function on that result. Note that $\sin3$ is itself an angle in radians now, and it lies in $[-1,1]$ as every sine value must:

   $\cos(0.1411200)=0.9900590$

   For small $\theta$, $\cos\theta\approx1-\tfrac{\theta^2}{2}=1-\tfrac{0.14112^2}{2}=1-0.00996=0.99004$ — matching to four decimals ✓.

5. 5

   State the answer.

   $\cos(\sin3)\approx0.99006$

6. 6

   Note the range of the whole composition. Because $\sin3$ can only lie in $[-1,1]$, the outer cosine is evaluated on $[-1,1]$, where it is at least $\cos1=0.5403$. So $\cos(\sin x)$ always lies in $[0.5403,1]$ for every real $x$ — it can never be negative. Getting a negative answer would be a sure sign of a mode or bracketing error.