Logarithms, trigonometry and numerical methods (Pure 3)
Same skills as Pure 2
Pure 3 reuses three Pure-2 areas exactly:
- logarithms (3.2) — the log laws with $e^x$ and $\ln x$.
- trigonometry (3.3) — $\sec^2\theta \equiv 1 + \tan^2\theta$, $\csc^2\theta \equiv 1 + \cot^2\theta$, the compound- and double-angle formulae, and the $R$-form of $a\sin\theta + b\cos\theta$.
- numerical methods (3.6) — a sign change to trap a root, then an iterative formula $x_{n+1} = F(x_n)$.
Practice
Using sec²θ ≡ 1 + tan²θ, if tan θ = 3, what is sec²θ?
sec²θ = 1 + 3² = 1 + 9 = 10.
Practice
What is ln(e³)?
ln and e are inverses, so ln(e³) = 3.
Practice
A numerical root is found by a sign change, then repeatedly applying:
Iteration applies x_{n+1} = F(x_n) until the values converge to the root.
You've got it
Key idea
- 3.2 logs, 3.3 trig identities/angle formulae, 3.6 numerical methods = the Pure 2 skills
- use $\sec^2\theta \equiv 1 + \tan^2\theta$ and the R-formula as before
- find roots by a sign change then iteration $x_{n+1} = F(x_n)$