Find derivatives and integrals of $\sin$, $\cos$ and $\tan$ (with linear inside arguments) and apply them to model simple harmonic and periodic motion

What this dot point is asking

NESA wants you to differentiate and integrate the three standard trigonometric functions, including with linear inside arguments such as $\sin(k x + c)$, and apply this calculus to model periodic phenomena (tides, oscillating springs, biological cycles).

The answer

Derivatives

The angle is always measured in radians. The derivatives are:

With a linear inside argument, apply the chain rule:

Integrals

With a linear inside argument, divide by the inside coefficient:

Modelling periodic motion

A function of the form

describes simple harmonic motion (or any sinusoidal cycle). The parameters are:

• IMATH_13 is the amplitude (half the peak-to-trough range).
• IMATH_14 is the angular frequency in radians per unit time. The period is $T = \frac{2 \pi}{\omega}$.
• IMATH_16 is the phase shift.
• IMATH_17 is the vertical shift (the centre of oscillation).

Differentiating gives the velocity, $y'(t) = A \omega \cos(\omega t + \phi)$, and differentiating again gives the acceleration, $y''(t) = -A \omega^2 \sin(\omega t + \phi) = -\omega^2 (y - D)$. The acceleration is proportional to the displacement from the centre and points back towards it.

Worked example: derivative of a composite

Differentiate $y = \sin^2 x$.

Rewrite as $y = (\sin x)^2$ and apply the chain rule with $u = \sin x$.

$\frac{dy}{dx} = 2 \sin x \cdot \cos x = \sin(2 x)$,

using the double-angle identity.

Worked example: integral via substitution

Evaluate $\int \sin x \cos x \, dx$.

Let $u = \sin x$, so $du = \cos x \, dx$.

$\int u \, du = \frac{u^2}{2} + C = \frac{\sin^2 x}{2} + C$.

Equivalently, you could rewrite using $\sin x \cos x = \frac{1}{2} \sin(2 x)$ and integrate directly, giving $-\frac{1}{4} \cos(2 x) + C$. The two forms differ by a constant.

Worked example: harmonic motion

A spring oscillates so that its displacement from equilibrium is $x(t) = 0.1 \cos(4 t)$ metres, with $t$ in seconds.

Velocity. $v(t) = x'(t) = -0.4 \sin(4 t)$ m/s.

Acceleration. $a(t) = v'(t) = -1.6 \cos(4 t) = -16 \, x(t)$ m/s$^2$.

The maximum speed is $0.4$ m/s, the period is $T = \frac{2 \pi}{4} = \frac{\pi}{2}$ seconds, and the acceleration is always directed back toward equilibrium with magnitude proportional to displacement.

Common traps

Working in degrees. 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.

Sign error on $\cos$. $\frac{d}{dx}(\cos x) = -\sin x$ and $\int \sin x \, dx = -\cos x + C$. Forgetting the minus is the single most common arithmetic slip.

Forgetting to divide by the inside coefficient when integrating. $\int \cos(3 x) \, dx = \frac{1}{3} \sin(3 x) + C$, not $\sin(3 x) + C$.

Treating $\sin^2 x$ like $\sin(x^2)$. $\sin^2 x = (\sin x)^2$ has derivative $2 \sin x \cos x$. $\sin(x^2)$ has derivative $2 x \cos(x^2)$.

Confusing period and angular frequency. In $\sin(\omega t)$, the angular frequency is $\omega$ and the period is $\frac{2 \pi}{\omega}$.

In one sentence

Trigonometric calculus differentiates and integrates $\sin$, $\cos$ and $\tan$ in radians, with chain-rule and reverse-chain-rule adjustments for linear inside arguments, and underpins the modelling of every periodic phenomenon in Maths Advanced.