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NSWMaths AdvancedQuick questions
Year 12: Calculus
Quick questions on Calculus of trigonometric functions: derivatives, integrals and harmonic motion modelling
10short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is derivatives?Show answer
The angle is always measured in radians. The derivatives are:
What is integrals?Show answer
$$\int \sin x \, dx = -\cos x + C$$
What is modelling periodic motion?Show answer
$$y(t) = A \sin(\omega t + \phi) + D$$
What is velocity?Show answer
$v(t) = x'(t) = -0.4 \sin(4 t)$ m/s.
What is acceleration?Show answer
$a(t) = v'(t) = -1.6 \cos(4 t) = -16 \, x(t)$ m/s$^2$.
What is working in degrees?Show answer
The derivative rules assume radians. If you differentiate $\sin x$ in degrees, you get $\frac{\pi}{180} \cos x$, which is not what NESA expects.
What is sign error on $\cos$?Show answer
$\frac{d}{dx}(\cos x) = -\sin x$ and $\int \sin x \, dx = -\cos x + C$. Forgetting the minus is the single most common arithmetic slip.
What is forgetting to divide by the inside coefficient when integrating?Show answer
$\int \cos(3 x) \, dx = \frac{1}{3} \sin(3 x) + C$, not $\sin(3 x) + C$.
What is treating $\sin^2 x$ like $\sin $?Show answer
$\sin^2 x = (\sin x)^2$ has derivative $2 \sin x \cos x$. $\sin(x^2)$ has derivative $2 x \cos(x^2)$.
What is confusing period and angular frequency?Show answer
In $\sin(\omega t)$, the angular frequency is $\omega$ and the period is $\frac{2 \pi}{\omega}$.