Calculus I · 2A · lesson

Derivatives of Sine and Cosine

Learn derivatives of sine and cosine with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

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Section overview

Trigonometric, exponential, and logarithmic functions

What this section is building

Learn derivatives of sine and cosine with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.

Notice

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

Decide

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

Avoid

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

Concept

Learning objectives

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

Trigonometric, Exponential, and Logarithmic Derivatives

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

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 $\ln x$ follows from its inverse relationship with $e^x$ ; 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.

**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 $\sin x$ , $\cos x$ , $-\sin x$ , and $-\cos x$ .

Angles must be measured in radians for the clean derivative formulas to hold. Radians connect arc length directly to angle, allowing the limit $\sin h/h\to1$ 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 $\sin h/h$ to approach $1$ . 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

\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

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

Use $\sin(x+h)=\sin x\cos h+\cos x\sin h$ :

\sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h}.

As $h\to0$ , the first quotient approaches $0$ and the second approaches $1$ . The limit is $\cos x$ . The cosine proof uses the angle-addition identity and yields $-\sin x$ .

Guided walkthrough

Differentiate a trigonometric polynomial

Differentiate

f(x)=3\sin x-4\cos x+x^3.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A tangent line to sine

At $x=\pi/3$ , $y=\sin x$ has point

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

and slope

\cos\frac\pi3=\frac12.

Thus the tangent line is

\boxed{y-\frac{\sqrt3}{2}=\frac12\left(x-\frac\pi3\right)}.

Interactive checksin-cos-01

Differentiate $4\sin x+2\cos x$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Differentiate sine to cosine and cosine to negative sine.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Modeling lab

Vertical velocity on a Ferris wheel

A rider's height is

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

meters, where one revolution takes $40$ seconds. Then

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

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

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

Interactive checktrig-extra-01

Differentiate $\cos x$ .

Your work stays on this device. No account or AI grader is used.

Show hint

The cosine derivative is negative sine.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkapp-ferris-01

For $h(t)=22-18\cos(\pi t/20)$ , find $h'(10)$ .

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

The Derivatives of

\sin x

and

\cos x

Explanation

Before the formulas

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

The familiar formulas require radians. Radian measure makes arc length directly proportional to angle, allowing the local ratio

\sin h/h

to approach

1

. 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

\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

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

Use $\sin(x+h)=\sin x\cos h+\cos x\sin h$ :

\sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h}.

As $h\to0$ , the first quotient approaches $0$ and the second approaches $1$ . The limit is $\cos x$ . The cosine proof uses the angle-addition identity and yields $-\sin x$ .

Guided walkthrough

Differentiate a trigonometric polynomial

Differentiate

f(x)=3\sin x-4\cos x+x^3.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

A tangent line to sine

At $x=\pi/3$ , $y=\sin x$ has point

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

and slope

\cos\frac\pi3=\frac12.

Thus the tangent line is

\boxed{y-\frac{\sqrt3}{2}=\frac12\left(x-\frac\pi3\right)}.

Interactive checksin-cos-01

Differentiate $4\sin x+2\cos x$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Differentiate sine to cosine and cosine to negative sine.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Modeling lab

Vertical velocity on a Ferris wheel

A rider's height is

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

meters, where one revolution takes $40$ seconds. Then

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

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

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

Interactive checktrig-extra-01

Differentiate $\cos x$ .

Your work stays on this device. No account or AI grader is used.

Show hint

The cosine derivative is negative sine.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkapp-ferris-01

For $h(t)=22-18\cos(\pi t/20)$ , find $h'(10)$ .

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Trigonometric, Exponential, and Logarithmic Derivatives

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

Related resources

Original instruction with traceable references.

Trigonometric, Exponential, and Logarithmic Derivatives

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

Related resources

Original instruction with traceable references.

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

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