Given that $\theta $ is small, show that \(\sec \theta \approx 1+\frac{1}{2}{{\theta }^{

Question

Given that $\theta $ is small, show that $\sec \theta \approx 1+\frac{1}{2}{{\theta }^{2}}$.[2] The diagram below shows a circle, centre $O$ and radius $r$, with points $A$ and $B$ on the circumference such that $\angle AOB=\theta $ radians, where $\theta $ is small. $AC$ is a tangent to the circle at $A$ and $OBC$ is a straight line. Show that the length of chord $AB$ can be approximated by $r\theta $.[2] Hence show that the perimeter of triangle $ABC$ can be approximated by $r\theta (a+b\theta )$, where $a$ and $b$ are constants to be determined.[4]

Accepted Answer

(a) $\sec\theta\approx1+\frac12\theta^2$. (b)(i) $AB\approx r\theta$. (b)(ii) The perimeter is $r\theta(2+\frac12\theta)$, so $a=2$ and $b=\frac12$.

2025 ASRJC P1 Q3

2025 ASRJC P1 Q3

Video Solution:

Solution:

Solution locked