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Calculus I · Unit 2A · lesson

Derivatives of Sine and Cosine

Concept

Learning objectives

Differentiate sine and cosine; understand how the fundamental trigonometric limits produce the rules.

Trigonometric, Exponential, and Logarithmic Derivatives

The Derivatives of sin⁡x\sin x and cos⁡x\cos x

Explanation

Before the formulas

Special-function derivatives become less mysterious in Derivatives of Sine and Cosine when you connect them to inverses and limits. The derivative of ln⁡x\ln x follows from its inverse relationship with exe^x; the derivative of tangent follows from sine, cosine, and the quotient rule. Learning these connections reduces memorization and provides recovery routes when a formula is forgotten.

Use a graph or a numerical point to check signs and scale. If the function is increasing rapidly, its derivative should be positive and large. If the function is undefined at an input, its derivative formula cannot rescue that input.

The slope pattern of sin x is exactly cos x. The slope pattern of cos x is -sin x. Matching heights on the derivative graph to slopes on the original graph makes the formulas visible.
Read this graph as text

Sine and cosine derivatives are phase relationships. The slope pattern of x is exactly x . The slope pattern of x is - x . Matching heights on the derivative graph to slopes on the original graph makes the formulas visible. In the upper panel, the dashed cosine curve is positive exactly where sine is increasing, zero at sine's peaks and troughs, and negative where sine is decreasing. In the lower panel, the dashed negative-sine curve plays the same role for cosine. The derivative formulas encode these slope patterns.

The visual uses labeled positions, solid and dashed line styles, and written descriptions so sine and cosine derivatives are phase relationships does not depend on color.

Why it matters: This figure turns two memorized formulas into graph-reading facts. It should be used alongside a unit-circle derivation, but it remains valuable afterward as a sign and phase check. Students can detect a missing minus sign in ( x)' by asking whether cosine is decreasing just to the right of zero.

Visual study

The slope pattern of sin x is exactly cos x. The slope pattern of cos x is -sin x. Matching heights on the derivative graph to slopes on the original graph makes the formulas visible.

Explanation

Trigonometric derivatives describe circular motion

Sine and cosine are not arbitrary special cases. They are coordinates of motion around a unit circle. As a point moves, the horizontal and vertical coordinates trade roles in a quarter-turn pattern, which is why differentiation cycles through sin⁡x\sin x, cos⁡x\cos x, −sin⁡x-\sin x, and −cos⁡x-\cos x.

Angles must be measured in radians for the clean derivative formulas to hold. Radians connect arc length directly to angle, allowing the limit sin⁡h/h→1\sin h/h\to1 to produce derivatives with no conversion constant.

Sine and cosine are not polynomial decorations. They encode periodic motion: rotations, waves, seasons, vibrations, and repeating signals. Their derivatives swap the two functions, with a negative sign appearing for cosine because cosine is decreasing at the origin while sine is increasing.

The familiar formulas require radians. Radian measure makes arc length directly proportional to angle, allowing the local ratio sin⁡h/h\sin h/h to approach 11. Degree measure inserts an unwanted conversion factor into every derivative.

Angles must be measured in radians for the standard derivative formulas.

Theorem

Sine and cosine derivatives

ddxsin⁡x=cos⁡x,ddxcos⁡x=−sin⁡x.\boxed{\frac{d}{dx}\sin x=\cos x}, \qquad \boxed{\frac{d}{dx}\cos x=-\sin x}.
Proof idea

Why the derivative of sine is cosine

Start from

sin⁡(x+h)−sin⁡xh.\frac{\sin(x+h)-\sin x}{h}.

Use sin⁡(x+h)=sin⁡xcos⁡h+cos⁡xsin⁡h\sin(x+h)=\sin x\cos h+\cos x\sin h:

sin⁡xcos⁡h−1h+cos⁡xsin⁡hh.\sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h}.

As h→0h\to0, the first quotient approaches 00 and the second approaches 11. The limit is cos⁡x\cos x. The cosine proof uses the angle-addition identity and yields −sin⁡x-\sin x.

Guided walkthrough

Differentiate a trigonometric polynomial

Differentiate

f(x)=3sin⁡x−4cos⁡x+x3.f(x)=3\sin x-4\cos x+x^3.
Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Worked example

A tangent line to sine

At x=π/3x=\pi/3, y=sin⁡xy=\sin x has point

(π3,32)\left(\frac\pi3,\frac{\sqrt3}{2}\right)

and slope

cos⁡π3=12.\cos\frac\pi3=\frac12.

Thus the tangent line is

y−32=12(x−π3).\boxed{y-\frac{\sqrt3}{2}=\frac12\left(x-\frac\pi3\right)}.
Interactive checksin-cos-01

Differentiate 4sin⁡x+2cos⁡x4\sin x+2\cos x.

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Show hint

Differentiate sine to cosine and cosine to negative sine.

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Submit an answer first. The hint is available now.

Modeling lab

Vertical velocity on a Ferris wheel

A rider's height is

h(t)=22−18cos⁡(πt20)h(t)=22-18\cos\left(\frac{\pi t}{20}\right)

meters, where one revolution takes 4040 seconds. Then

h′(t)=18π20sin⁡(πt20).h'(t)=\frac{18\pi}{20}\sin\left(\frac{\pi t}{20}\right).

At t=10t=10, the rider is halfway up the first ascent and

h′(10)=9π10 m/s.h'(10)=\frac{9\pi}{10}\text{ m/s}.

At the bottom and top, the derivative is zero because the rider's vertical direction reverses, even though the wheel never stops rotating.

Optional advanced note

Why radians are mathematically natural

For an angle hh in radians on the unit circle, the arc length is exactly hh. This geometric identity leads to sin⁡h/h→1\sin h/h\to1. If the angle is measured in degrees, the arc length is πh/180\pi h/180, and the derivative of sin⁡h\sin h acquires the factor π/180\pi/180. Radians are not a convention chosen to annoy students; they are the coordinate in which circular motion has unit local scale.

After the explanation

Use the section idea

Reading lens

Connect each derivative formula to the function's characteristic growth, periodicity, or inverse relationship.

Mental model

Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.

Decision

Identify the function family first, then check whether a composition requires the chain rule too.

Common trap

Using a power rule on an exponential or forgetting base and domain conditions for logarithms.

Check yourself

Can you distinguish a power, exponential, logarithmic, and trigonometric derivative at a glance?

Interactive checktrig-extra-01

Differentiate cos⁡x\cos x.

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Show hint

The cosine derivative is negative sine.

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Submit an answer first. The hint is available now.

Interactive checkapp-ferris-01

For h(t)=22−18cos⁡(πt/20)h(t)=22-18\cos(\pi t/20), find h′(10)h'(10).

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Show hint

Differentiate with the chain rule and use sin⁡(π/2)=1\sin(\pi/2)=1.

Attempt once to unlock the solution

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Original instruction with traceable references.

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