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:

• $A$ is the amplitude (half the peak-to-trough range).
• $\omega$ is the angular frequency in radians per unit time. The period is $T = \frac{2 \pi}{\omega}$.
• $\phi$ is the phase shift.
• $D$ 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.

Why radians are compulsory

The derivative rules $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$ only hold when $x$ is measured in radians, because they rest on the limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$, which is true only in radian measure. If your calculator is in degree mode the geometry of the limit changes and every derivative is off by a factor of $\frac{\pi}{180}$. Always set angles in radians for any calculus involving trigonometric functions, and convert any degree information in the question to radians before differentiating or integrating.

Why the derivative of $\sin x$ is $\cos x$

The rule $\frac{d}{dx}\sin x = \cos x$ is worth seeing, not just memorising. The derivative of a function is its gradient at each point, so if we read the gradient of $\sin x$ off the graph and plot those gradient values, we should recover $\cos x$. We do.

Stage 1, plot $y = \sin x$. Draw one full cycle of $\sin x$ from $0$ to $2\pi$: up to a peak at $\frac{\pi}{2}$, back through zero at $\pi$, down to a trough at $\frac{3\pi}{2}$, and back to zero at $2\pi$.

Stage 2, measure the gradient at sample points. Draw the tangent at a few key points and read its gradient. At $x = 0$ the curve rises steeply, gradient $+1$; at the peak $x = \frac{\pi}{2}$ the tangent is flat, gradient $0$; at $x = \pi$ the curve falls steeply, gradient $-1$; at the trough it is flat again; at $2\pi$ it is rising, gradient $+1$.

Stage 3, plot each gradient against $x$. Take those gradient values, $1, 0, -1, 0, 1$, and plot them at the same $x$-positions. The points sit exactly where $\cos x$ would: $\cos 0 = 1$, $\cos\frac{\pi}{2} = 0$, $\cos\pi = -1$, and so on.

Stage 4, the gradient function is $\cos x$. Join the gradient points and the curve $y = \cos x$ appears. The gradient of $\sin x$ at every point is $\cos x$, which is exactly what $\frac{d}{dx}\sin x = \cos x$ says. The same picture, shifted, shows $\frac{d}{dx}\cos x = -\sin x$.

Useful identities for integration

Some trigonometric integrals have no direct antiderivative until you rewrite them with an identity. The most useful in Maths Advanced are the double-angle forms $\sin^2 x = \frac{1 - \cos 2x}{2}$ and $\cos^2 x = \frac{1 + \cos 2x}{2}$. These convert a squared trig function (which you cannot integrate directly) into a constant plus a cosine of a double angle (which you can). For example, $\int \cos^2 x\,dx = \int \frac{1 + \cos 2x}{2}\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$. Recognising when to deploy an identity is a key marker of fluency.

How exam questions ask about trigonometric calculus

• "Differentiate" a trig expression. Apply the standard derivative and the chain-rule factor $k$ for a linear inside argument; watch the minus sign on $\cos$.
• "Find $\int \ldots \, dx$" or "evaluate $\int_a^b \ldots \, dx$". Use the standard integral, divide by the inside coefficient, add $+C$ for an indefinite integral, and substitute the limits for a definite one.
• "Find the rate at which ... is changing." A modelling question: differentiate the given function and substitute the value, exact then with units.
• "Find the maximum / minimum value" or "the period / amplitude." Read the parameters of $A\sin(\omega t + \phi) + D$: amplitude $A$, period $\frac{2\pi}{\omega}$, centre $D$; the extreme values are $D \pm A$.
• "Show that the motion is simple harmonic." Differentiate twice and show $y'' = -\omega^2(y - D)$: acceleration proportional to displacement and directed back to the centre.
• An integral of $\sin^2 x$ or $\cos^2 x$. Rewrite with the double-angle identity first; the squared form has no direct antiderivative.