Evaluate tan(7 degrees) to four decimal places.

Step-by-step solution

1. 1

   Convert the angle to radians. Multiplying by $\tfrac{\pi}{180}$:

   $7^\circ=0.1221730\ \text{rad}$

2. 2

   Use the small-angle behaviour to predict the answer. For small $x$ in radians, $\tan x\approx x$, so the value should be a little above $0.1222$. The correction term is cubic:

   $\tan x\approx x+\frac{x^3}{3}$

3. 3

   Apply the series. With $x=0.1221730$, $x^3=0.00182357$, so

   $\tan x\approx 0.1221730+0.0006079=0.1227809$

   Already accurate to five decimals, since the next term is of order $x^5\approx 3\times 10^{-5}$ times $\tfrac{2}{15}$.

4. 4

   Cross-check with the quotient definition. $\tan\theta=\tfrac{\sin\theta}{\cos\theta}$, and with $\sin 7^\circ\approx 0.1218693$, $\cos 7^\circ\approx 0.9925462$:

   $\frac{0.1218693}{0.9925462}\approx 0.1227846$

5. 5

   Round and confirm. To four decimal places

   $\tan(7^\circ)\approx 0.1228$

   A direct evaluation gives $0.12278456$ $\checkmark$. Note it exceeds $\sin 7^\circ$, as it must, because dividing by $\cos 7^\circ<1$ increases the value.