Trigonometry and Circular Functions
Core concepts
Move fluently between degrees and radians; calculus and circular functions require radians. Know the unit circle, exact values for multiples of $\pi/6$, $\pi/4$ and $\pi/3$, and the signs of sine, cosine and tangent in each quadrant. Sketch $y=a\sin(b(x-c))+d$ and the corresponding cosine and tangent graphs, identifying amplitude, period ($2\pi/b$ for sine and cosine, $\pi/b$ for tangent), phase shift and vertical translation. Solve trigonometric equations over a stated domain, generating all solutions using periodicity and symmetry. Apply the identities $\sin^2\theta+\cos^2\theta=1$, $\tan\theta=\sin\theta/\cos\theta$ and the double-angle formulas. Differentiate and integrate sine and cosine, remembering the chain rule and radian requirement.
Traps
- Missing solutions because only the principal value was found.
- Using degrees when differentiating or integrating.
- Confusing amplitude with vertical translation, or period with phase shift.
- Forgetting the negative sign when integrating $\sin$ or differentiating $\cos$.
- Dividing by $\cos\theta$ and losing solutions where $\cos\theta=0$.
Worked example
Solve $2\sin(2x)=1$ for $0\le x\le 2\pi$. Then $\sin(2x)=\tfrac12$, so $2x=\pi/6,,5\pi/6,,13\pi/6,,17\pi/6,\dots$ within $[0,4\pi]$, giving $x=\pi/12,,5\pi/12,,13\pi/12,,17\pi/12$.
Rule of thumb
When solving, find the general solution first, then restrict to the required domain by adding or subtracting whole periods.
Exponential and Logarithmic Functions
Core concepts
Exponential functions $y=a^x$ (with $a>0$, $a\neq 1$) have domain all reals, range $y>0$, a horizontal asymptote at $y=0$, and are strictly increasing or decreasing. The natural base $e$ gives $y=e^x$, whose derivative is itself. Logarithms are inverses of exponentials: $y=\log_a x$ has domain $x>0$ and range all reals. Know the log laws: $\log(xy)=\log x+\log y$, $\log(x/y)=\log x-\log y$, $\log(x^n)=n\log x$, and change of base. Differentiate $e^{f(x)}$ and $\ln f(x)$ using the chain rule: $\frac{d}{dx}e^{f(x)}=f'(x)e^{f(x)}$ and $\frac{d}{dx}\ln f(x)=f'(x)/f(x)$. Integrate to match, and apply these models to growth, decay and half-life problems.
Traps
- Applying log laws to sums: $\log(x+y)\neq \log x+\log y$.
- Forgetting the domain restriction $x>0$ for logarithms.
- Losing the chain-rule factor when differentiating $e^{kx}$ or $\ln(kx)$.
- Solving $e^x=-3$ and reporting a solution when none exists.
- Mixing up growth and decay rates in modelling.
Worked example
Differentiate $y=e^{3x}\ln x$. By the product rule, $y'=3e^{3x}\ln x+e^{3x}\cdot\frac1x=e^{3x}\left(3\ln x+\frac1x\right)$.
Rule of thumb
To solve an exponential equation, isolate the exponential and take logarithms of both sides; to solve a log equation, combine logs and convert to exponential form, then check the domain.