June 2022 Paper 3 Q9

9

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Approximating the sine function” are reproduced below; the line numbers are those printed on the Insert.

Line 1
Small angles

Lines 2–5
For a small angle \(x\) radians, the approximation \(\sin x \approx x\) is valid. The curve \(y = \sin x\) and the straight line \(y = x\) are shown in Fig. C1.1. Fig. C1.2 shows the curve \(y = x - \sin x\). Inspection of the graphs suggests that \(x\) is a reasonable approximation for \(\sin x\) for \(-0.5 \leqslant x \leqslant 0.5\) and also that \(y = x\) has the same gradient as \(y = \sin x\) when \(x = 0\).

Show that \(y = x\) has the same gradient as \(y = \sin x\) when \(x = 0\), as stated in line 5. [2]