Trigonometric derivatives
Connect sine and cosine to their slopes, then apply the differentiation rules.
The idea
Sine rises, flattens, falls, and repeats. Its tangent slope follows the same repeating pattern as cosine. The paired graphs below connect a function’s height with its rate of change.
Rules
Sine and cosine
\[\frac{d}{dx}\sin x=\cos x,\qquad \frac{d}{dx}\cos x=-\sin x.\]Both rules hold for every real input, with the angle in radians.
Derivations (optional)
Use the angle-addition identities and the earlier limits
\[\lim_{h\to0}\frac{\sin h}{h}=1,\qquad \lim_{h\to0}\frac{\cos h-1}{h}=0.\]The second follows from \(\frac{\cos h-1}{h}=-h\frac{1-\cos h}{h^2}\) and the previously established cosine limit. Then
\[\begin{aligned} \frac{d}{dx}\sin x &=\lim_{h\to0}\frac{\sin(x+h)-\sin x}{h}\\ &=\lim_{h\to0}\left[\sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h}\right]\\ &=\sin x\cdot0+\cos x\cdot1=\cos x. \end{aligned}\]Similarly,
\[\begin{aligned} \frac{d}{dx}\cos x &=\lim_{h\to0}\frac{\cos(x+h)-\cos x}{h}\\ &=\lim_{h\to0}\left[\cos x\frac{\cos h-1}{h}-\sin x\frac{\sin h}{h}\right]\\ &=\cos x\cdot0-\sin x\cdot1=-\sin x. \end{aligned}\]At \(x=\pi/2\), sine is at a crest and its slope is \(0\). At \(x=\pi\), it crosses zero while falling, and its slope is \(-1\). The derivative tracks change, not height.
The other four functions
Their derivatives follow from the quotient rule.
Rules
Tangent, cotangent, secant, and cosecant
\[\frac{d}{dx}\tan x=\sec^2 x,\qquad \frac{d}{dx}\cot x=-\csc^2 x.\] \[\frac{d}{dx}\sec x=\sec x\tan x,\qquad \frac{d}{dx}\csc x=-\csc x\cot x.\]Use these only where the original functions are defined: \(\cos x\ne0\) for tangent and secant; \(\sin x\ne0\) for cotangent and cosecant.
For example, the quotient rule explains the tangent rule:
\[\frac{d}{dx}\tan x=\frac{\cos x\cos x-\sin x(-\sin x)}{\cos^2x}=\frac{1}{\cos^2x}=\sec^2x.\]Worked examples
Example 1
An angle that also changes
For \(y=\sin(x^2)\), the inner angle is \(x^2\). Use the chain rule:
\[\frac{dy}{dx}=\cos(x^2)\cdot2x.\]Differentiate sine at the current angle, then multiply by the rate at which that angle changes.
Example 2
A quotient and an identity
For \(q(x)=\frac{\cos x}{1+\sin x}\), where \(1+\sin x\ne0\),
\[\begin{aligned} q^{\prime}(x)&=\frac{(-\sin x)(1+\sin x)-\cos x\cos x}{(1+\sin x)^2}\\ &=\frac{-\sin x-(\sin^2x+\cos^2x)}{(1+\sin x)^2}\\ &=-\frac{1}{1+\sin x}. \end{aligned}\]The identity \(\sin^2x+\cos^2x=1\) makes the simplification possible. The original domain restriction remains.
Example 3
Position and velocity in periodic motion
An object has position \(s(t)=2\cos(3t)\) metres, with time in seconds and the angle \(3t\) in radians. Its instantaneous velocity is
\[s^{\prime}(t)=-2\sin(3t)\cdot3=-6\sin(3t)\,\mathrm{m/s}.\]At \(t=\pi/6\) seconds, the object has position \(0\) and velocity \(-6\,\mathrm{m/s}\). It is passing through the origin in the negative direction, rather than stopping there.
The coefficient \(2\) is the amplitude: maximum displacement from the centre. The coefficient \(3\) is the angular frequency, describing how quickly the angle changes in radians per second.