Logarithms, trigonometry and numerical methods (Pure 3)

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.

A line r = a + t b: start at a, then slide by t lots of the direction b

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.

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.

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.

You've got it

Write from the solution x equals 5

• Solve e to the 2x = e to the 10.