Evaluate cos(7 degrees) to four decimal places.

Step-by-step solution

1. 1

   Recognise that $7^\circ$ is not a special angle. It is not one of $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$ or their reflections, so there is no simple exact form and the answer must be numerical.

2. 2

   Convert to radians, which is what the series and most software expect. Multiply by $\tfrac{\pi}{180}$:

   $7^\circ=7\cdot\frac{\pi}{180}\approx 0.1221730\ \text{rad}$

3. 3

   Predict the size before computing. Since $7^\circ$ is close to $0^\circ$ and $\cos 0=1$, the answer must be just below $1$. Cosine is also decreasing on $[0^\circ,90^\circ]$, so $\cos 7^\circ<\cos 0^\circ=1$.

4. 4

   Estimate with the Maclaurin series. With $x=0.1221730$:

   $\cos x\approx 1-\frac{x^2}{2}+\frac{x^4}{24}=1-0.0074631+0.0000093=0.9925462$

   Three terms already pin down four decimals because $x$ is small.

5. 5

   Round and confirm. To four decimal places

   $\cos(7^\circ)\approx 0.9925$

   A direct evaluation gives $0.99254615$, matching the series estimate to seven decimals $\checkmark$.