Book A Free Math Class

# Differentiation of Trigonometric Functions — Formulas & Rules

[#Trigonometry](/content/tag/trigonometry/index.html)

## TL;DR

The differentiation of trigonometric functions gives the six core rules:

1. \( \frac{d}{dx}\sin x = \cos x \)  
2. \( \frac{d}{dx}\cos x = -\sin x \)  
3. \( \frac{d}{dx}\tan x = \sec^2 x \)  
4. \( \frac{d}{dx}\cot x = -\csc^2 x \)  
5. \( \frac{d}{dx}\sec x = \sec x \tan x \)  
6. \( \frac{d}{dx}\csc x = -\csc x \cot x \)

All six follow from the sine and cosine derivatives via the quotient rule. This article proves them from first principles and shows where students slip.

## A Swing With No Ending

Galileo timed a chandelier swinging during mass in 1583 and noticed something odd — the swing took the same time whether the arc was wide or narrow. That observation became calculus's first physical example of _position → velocity → acceleration_ under sine and cosine. The reason the derivative of \( \, \sin x \) is \( \cos x \) isn't a notation trick. It's that velocity always sits one quarter-period ahead of position in any oscillating system.

## What Is Differentiation of Trigonometric Functions?

**Differentiation of trigonometric functions** is the process of finding the derivative — the instantaneous rate of change — of \( \, \sin x \, \), \( \, \cos x \, \), \( \, \tan x \, \), and their reciprocal partners. The derivatives come in clean pairs: each function's derivative is closely related to another trig function (or its negative). All six rules are proved from two foundational limits and the quotient rule.

## The Six Derivative Rules

| Function              | Derivative            |
|-----------------------|-----------------------|
| \( \sin x \)       | \( \cos x \)       |
| \( \cos x \)      | \( -\sin x \)      |
| \( \tan x \)      | \( \sec^2 x \)     |
| \( \cot x \)      | \( -\csc^2 x \)    |
| \( \sec x \)      | \( \sec x \tan x \)|
| \( \csc x \)      | \( -\csc x \cot x \)|

**Pattern to lock in.** Every "co-" function (cosine, cotangent, cosecant) has a _negative_ sign in its derivative. The three "non-co" functions don't. That single rule recovers half the table on exam day if memory fails.

These rules assume \( x \) is measured in **radians**. If \( x \) is in degrees, every derivative picks up a factor of \( \frac{\pi}{180} \) — which is why no calculus textbook works in degrees. We come back to this in the mistakes section.

## Proof From First Principles — Derivative of \( \sin x \)

Starting from the limit definition:

\[ \frac{d}{dx}\sin x = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h} \]

Apply the sum identity \( \sin(x+h) = \sin x \cos h + \cos x \sin h \):

\[ = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h} \]

\[ = \lim_{h \to 0}\left[\sin x \cdot \frac{\cos h - 1}{h} + \cos x \cdot \frac{\sin h}{h}\right] \]

Two foundational limits do the rest:

- \( \lim_{h \to 0}\frac{\sin h}{h} = 1, \)  
- \( \lim_{h \to 0}\frac{\cos h - 1}{h} = 0 \)

So:

\[ \frac{d}{dx}\sin x = \sin x \cdot 0 + \cos x \cdot 1 = \cos x \]

The proof for \( \cos x \) is mechanically identical with \( \cos(x+h) \) in place of \( \sin(x+h) \). The other four derivatives — \( \tan, \cot, \sec, \csc \) — follow by writing each as a sine/cosine quotient and applying the quotient rule.

### Quick derivation of \( \tan x \)

\[ \frac{d}{dx}\tan x = \frac{d}{dx}\left(\frac{\sin x}{\cos x}\right) = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \sec^2 x \]

The numerator collapses by the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). That's why the trig derivatives are so tidy — Pythagoras is doing background work the whole time.

## The Chain Rule for Trigonometric Functions

When the argument is a function \( u(x) \), each derivative picks up \( u^{\prime}(x) \):

\[ \frac{d}{dx}\sin(u) = \cos(u) \cdot u^{\prime} \qquad \frac{d}{dx}\cos(u) = -\sin(u) \cdot u^{\prime} \]

\[ \frac{d}{dx}\tan(u) = \sec^2(u) \cdot u^{\prime} \qquad \frac{d}{dx}\sec(u) = \sec(u)\tan(u) \cdot u^{\prime} \]

The chain rule is where most exam mistakes happen — and it's also where most of the action is. Almost every physics or engineering derivative involves \( \sin(\omega t) \) or \( \cos(\omega t) \) — the chain rule pulls the angular-frequency \( \omega \) out front.

## Three Worked Examples — Quick, Standard, Stretch

### Quick

**Differentiate \( f(x)=3\sin x+2\cos x \)** .

By linearity:

\[ f^{\prime}(x)=3\cos x+2\cdot(-\sin x)=3\cos x-2\sin x \]

Done in one line. The negative sign on cosine is the only thing to watch.

### Where Students Lose the Mark — A Worked Standard Example

**Differentiate \( g(x)=\sin(3x^2) \)** .

_The wrong path._ A student writes:

\[ g^{\prime}(x)=\cos(3x^2) \text{❌} \]

They've remembered "the derivative of sine is cosine" and stopped there. The argument \( 3x^2 \) wasn't \( x \), so the chain rule applies — but it got skipped.

**Sanity check.** At \( x=0, \) this answer gives \( g^{\prime}(0)=\cos(0)=1 \). But \( g(x)=\sin(3x^2) \) is even (symmetric about the y-axis) — its derivative must be odd, and an odd function must satisfy \( g^{\prime}(0)=0 \). The answer 1 contradicts that. Something's missing.

_The correct path._ Apply the chain rule. Let \( u=3x^2, \) so \( u^{\prime}=6x \):

\[ g^{\prime}(x)=\cos(3x^2)\cdot\frac{d}{dx}(3x^2)=6x\cos(3x^2) \]

### Stretch

**A particle's position at time \( t \) seconds is \( s(t)=4\sin(\frac{2\pi t}{5}) \) metres. Find its velocity and the maximum speed.**

Velocity is \( s^{\prime}(t) \). With \( u=\frac{2\pi t}{5}, \) \( u^{\prime}=\frac{2\pi}{5} \):

\[ v(t)=4\cdot\cos\left(\frac{2\pi t}{5}\right)\cdot\frac{2\pi}{5} = \frac{8\pi}{5}\cos\left(\frac{2\pi t}{5}\right) \]

Maximum speed is the amplitude of the cosine — the term in front:

\[ v_{\max} = \frac{8\pi}{5} \approx 5.03 \, \text{m/s} \]

## Where These Derivatives Show Up in the Real World

The trig derivatives aren't just calculus furniture — they're how every oscillating system in physics gets analysed.

- **Simple harmonic motion.** Springs, pendulums, and tuning forks all satisfy \( s^{\prime\prime}=-\omega^2 s \) — and the only functions that solve that equation are \( \sin(\omega t) \) and \( \cos(\omega t) \). The reason the [Tacoma Narrows Bridge collapsed in 1940](https://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)) was that wind-induced vibrations matched the bridge's natural frequency — a calamity diagnosed afterward using exactly these derivatives.

- **AC electrical circuits.** Voltage in your wall outlet is \( V(t)=V_0\sin(2\pi\cdot60t) \) (in the US, McKinney TX included). Current through a capacitor is the _derivative_ of voltage — a cosine wave. That phase shift of 90° between voltage and current is just \( \frac{d}{dt}\sin = \cos \).

- **Sound and music.** Every musical note is a superposition of sine waves. The derivative — the rate at which air pressure changes — is what your eardrum actually senses.

- **Robotics and animation.** When an animated character bobs up and down, the animator usually scripts position as a sine wave; the velocity (used for momentum-based blending) is the cosine derivative.

## Tripping Points to Avoid

Four mistakes account for nearly every lost mark on this topic.

### **Mistake 1: Forgetting the negative sign on the "co-" derivatives**

**Where it slips in:** Anywhere \( \cos, \cot, \text{or} \csc \) appears in a longer expression.

### **Mistake 2: Skipping the chain rule when the argument isn't \( x \)**

**Where it slips in:** Composite arguments like \( \sin(3x^2) \), \( \cos(x^3) \), \( \tan(\ln x) \) — exactly the Standard example above.

### **Mistake 3: Working in degrees instead of radians**

**Where it slips in:** Calculator-heavy problems where the student forgets to switch the mode.

### **Mistake 4: Confusing the derivative of \( \sec x \) with \( \sec^2 x \)**

**Where it slips in:** Tangent and secant problems mixing up which one yields which derivative.

## Key Takeaways

- The **differentiation of trigonometric functions** rests on \( \frac{d}{dx}\sin x = \cos x \) and \( \frac{d}{dx}\cos x = -\sin x \) — everything else is the quotient rule.
- "Co-" functions (cos, cot, csc) get a negative sign; the other three don't.
- The chain rule isn't optional when the argument is anything other than \( x \) alone.
- All rules assume radians — degrees introduce a \( \frac{\pi}{180} \) factor that's silently lost on calculators.

## Try It Yourself — Three Problems

Differentiate the following without looking back at the table:

1. \( f(x)=\cos(5x) \)
2. \( g(x)=\tan(x^2+1) \)
3. \( h(x)=x\sin x \) (this one needs the product rule on top of the trig derivative)