Find the exact value of $\sec^{-1} 5$ (decimal answer).
I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ and used the Pythagorean Theorem to find the other side, which is $2\sqrt{6}$. The answer is $1.37$.
Thanks.
1 Answer
$\begingroup$You won't find the exact decimal answer. Alpha gives $1.369438406004565827776196139422128031858546618285324524230221\dots $ radians. If you need to compute it without a calculator that has inverse trig functions, you are in for some work. Probably the easiest approach is $\arccos \frac 15 = \frac \pi 2 - \arcsin \frac 15$, then use $\sin x \approx x-\frac {x^3}6$ to approximate $x$. An iteration $(x_0=\frac 15, x_{i+1}=\frac 15+\frac 16x_i^3$)quickly converges to $x \approx 0.201361$, giving $\arccos \frac 15 \approx 1.369436$
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