Logarithms, trigonometry and numerical methods (Pure 3)
| English | Chinese | Pinyin |
|---|---|---|
| logarithm | 对数 | duì shù |
| exponential equation | 指数方程 | zhǐ shù fāng chéng |
| decay | 衰减 | shuāi jiǎn |
| linearise | 线性化 | xiàn xìng huà |
| convergence | 收敛 | shōu liǎn |
Building on what you know
- Pure 3 doesn't introduce entirely new ideas — it deepens and extends the skills from Pure 2.
- Logarithms 对数, trigonometry, and numerical methods return with harder problems and richer applications.
Logarithms revisited
- The log laws with $e^x$ and $\ln x$ return in harder contexts: solving exponential equations 指数方程, modelling growth and decay 衰减, and linearising 线性化 data.
- Key skill: when the unknown is in the power, take logs of both sides.
Worked example. Solve $5e^{2x} = 30$. Divide: $e^{2x} = 6$. Take $\ln$: $2x = \ln 6$, so $x = \dfrac{\ln 6}{2} \approx 0.896$.
A line r = a + t b: start at a, then slide by t lots of the direction b
What is ln(e³)?
ln and e are inverses, so ln(e³) = 3.
Solve 5e^(2x) = 30 for x (2 dp).
e^(2x) = 6, so 2x = ln 6, x = (ln 6)/2 ≈ 0.896 ≈ 0.90.
Trigonometry revisited
- $\sec^2\theta \equiv 1 + \tan^2\theta$, $\csc^2\theta \equiv 1 + \cot^2\theta$.
- The compound- and double-angle formulae, and the $R$-form of $a\sin\theta + b\cos\theta$.
- These identities are essential for integration and solving trig equations.
Don't mix up the identities. $\sec^2\theta \equiv 1 + \tan^2\theta$ (plus sign), not $1 - \tan^2\theta$. The sign comes from dividing $\sin^2\theta + \cos^2\theta \equiv 1$ by $\cos^2\theta$.
The unit circle
(cos θ, sin θ)
The R-formula rewrites a sin θ + b cos θ as one wave — and it all lives on this circle.
Using sec²θ ≡ 1 + tan²θ, if tan θ = 3, what is sec²θ?
sec²θ = 1 + 3² = 1 + 9 = 10.
sec²θ ≡ 1 − tan²θ.
The correct identity is sec²θ ≡ 1 + tan²θ (plus, not minus).
Numerical methods revisited
- A sign change to trap a root, then an iterative formula $x_{n+1} = F(x_n)$.
- Convergence 收敛 requires $|F'(x)| < 1$ near the root.
allow_no_figure: Review lesson consolidating Pure 2 skills (logarithms, trigonometry, numerical methods) — no new visual concepts introduced.
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.
Worked example — R-formula application
- Express $3\sin\theta + 4\cos\theta$ in the form $R\sin(\theta + \alpha)$.
- $R = \sqrt{3^2 + 4^2} = 5$. $\tan\alpha = \dfrac{4}{3} \Rightarrow \alpha \approx 53.1^{\circ}$.
- So $3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.1^{\circ})$.
- Maximum value is $5$ (when $\sin(\theta + 53.1^{\circ}) = 1$).
- Plotting $\ln y$ against $x$ gives a line whose gradient and intercept find the constants (using indices and inverse functions); know the reciprocal trig functions secant, cosecant, cotangent; and iteration $x_{n+1}=F(x_n)$ shows convergence to a root.
For 3 sin θ + 4 cos θ = R sin(θ + α), what is R?
R = √(3² + 4²) = √25 = 5.
You've got it
- 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 $x_{n+1} = F(x_n)$