f′Field Guide
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Derivatives of Trigonometric Functions

Six formulas, two limits, and a cycle of four.

0 / 10 problems mastered
sin and cos
(sin⁡x)′=cos⁡x(\sin x)'=\cos x
(cos⁡x)′=−sin⁡x(\cos x)'=-\sin x
tan and cot
(tan⁡x)′=sec⁡2x(\tan x)'=\sec^2x
(cot⁡x)′=−csc⁡2x(\cot x)'=-\csc^2x
sec and csc
(sec⁡x)′=sec⁡xtan⁡x(\sec x)'=\sec x\tan x
(csc⁡x)′=−csc⁡xcot⁡x(\csc x)'=-\csc x\cot x
Memory trick

Every “co-” function (cos, cot, csc) gets a minus sign. tan ↔ sec² and cot ↔ csc² pair up; sec and csc reproduce themselves.

Trouble remembering these six? See Memory tricks: learn 3, get the other 3 with the “co” rule, then drill them with flashcards.

The slope of sin x is cos x

Lab 5a · drag on the graph
x (radians)–
slope of sin at x–
cos x–

The two special limits

Limit 1
lim⁡θ→0sin⁡θθ=1\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1

θ must be in radians. The argument of sin must match the denominator.

Limit 2
lim⁡θ→0cos⁡θ−1θ=0\lim_{\theta\to0}\frac{\cos\theta-1}{\theta}=0

Why sin θ / θ → 1

Lab 5b · shrink the angle
arc length θ–
vertical side sin θ–
sin θ / θ–

On a unit circle the arc has length θ and the pink segment has length sin θ. As θ shrinks, the arc straightens out and the two lengths become equal, so their ratio → 1.

Using the limit: make the denominator match the angle. Multiply and divide by what you need:

lim⁡x→0sin⁡7x4x=74lim⁡x→0sin⁡7x7x=74⋅1=74\lim_{x\to0}\frac{\sin 7x}{4x}=\frac74\lim_{x\to0}\frac{\sin7x}{7x}=\frac74\cdot1=\frac74

Higher derivatives of sin cycle every 4: sin⁡→cos⁡→−sin⁡→−cos⁡→sin⁡\sin\to\cos\to-\sin\to-\cos\to\sin. To find dndxnsin⁡x\frac{d^{n}}{dx^{n}}\sin x, divide nn by 4 and use the remainder.

Worked examples

Example · simplify after the quotient rule

Differentiate f(θ)=sin⁡θ1+cos⁡θf(\theta)=\dfrac{\sin\theta}{1+\cos\theta}. §3.3 #13

  1. f′=(1+cos⁡θ)cos⁡θ−sin⁡θ(−sin⁡θ)(1+cos⁡θ)2=cos⁡θ+cos⁡2θ+sin⁡2θ(1+cos⁡θ)2f'=\dfrac{(1+\cos\theta)\cos\theta-\sin\theta(-\sin\theta)}{(1+\cos\theta)^2}=\dfrac{\cos\theta+\cos^2\theta+\sin^2\theta}{(1+\cos\theta)^2}
  2. Use sin⁡2+cos⁡2=1\sin^2+\cos^2=1. Numerator =cos⁡θ+1=\cos\theta+1.
  3. f′(θ)=1+cos⁡θ(1+cos⁡θ)2=11+cos⁡θf'(\theta)=\dfrac{1+\cos\theta}{(1+\cos\theta)^2}=\dfrac{1}{1+\cos\theta}
Example · the 4-cycle

Find d27dx27cos⁡x\dfrac{d^{27}}{dx^{27}}\cos x.

  1. cos cycle: cos⁡→−sin⁡→−cos⁡→sin⁡→cos⁡\cos\to-\sin\to-\cos\to\sin\to\cos (period 4).
  2. 27=4⋅6+327=4\cdot6+3, so it equals the 3rd derivative.
  3. Answer: sin⁡x\sin x.

Practice

§3.3 #1
Differentiate f(x)=3sin⁡x−2cos⁡xf(x)=3\sin x-2\cos x.
§3.3 #5
Differentiate h(θ)=θ2sin⁡θh(\theta)=\theta^2\sin\theta.
§3.3 #7
Differentiate y=sec⁡θtan⁡θy=\sec\theta\tan\theta.
§3.3 #14
Differentiate y=cos⁡x1−sin⁡xy=\dfrac{\cos x}{1-\sin x}.
§3.3 #27
Find the tangent line to y=sin⁡x+cos⁡xy=\sin x+\cos x at (0,1)(0,1).
§3.3 #39
For what xx does f(x)=x+2sin⁡xf(x)=x+2\sin x have a horizontal tangent?
§3.3 #41
A mass on a spring has position x(t)=8sin⁡tx(t)=8\sin t cm. Find velocity and acceleration, and the state at t=2π/3t=2\pi/3. Which way is it moving?
§3.3 · limits
Find lim⁡t→0tan⁡6tsin⁡2t\displaystyle\lim_{t\to0}\frac{\tan6t}{\sin2t}.
§3.3 · limits
Find lim⁡x→0sin⁡3x5x3−4x\displaystyle\lim_{x\to0}\frac{\sin3x}{5x^3-4x}.
§3.3 · 4-cycle
Find d99dx99sin⁡x\dfrac{d^{99}}{dx^{99}}\sin x.

Traps

Degrees

All these formulas need radians. In degrees, (sin⁡x)′=π180cos⁡x(\sin x)'=\frac{\pi}{180}\cos x.

sin 7x / 4x → 7/4, not 1

The limit is 1 only when the angle and the denominator are identical.

Missing minus signs

cos, cot, csc all give negatives. Say “co- means minus” as you write.