PHYSICS TEACHING / CONCEPT GUIDE
Differentiating Trigonometric Functions
The derivatives of sine and cosine are not isolated rules to memorize—they form a mathematical cycle you can trace from limits and first principles.
This guide does more than state that the derivative of sine is cosine and the derivative of cosine is negative sine. It starts with two key limits and the definition of a derivative, then connects the resulting cycle to simple harmonic motion.
- Why physics needs trigonometric derivatives
- Simple harmonic motion commonly uses sine and cosine for position, velocity, and acceleration
- Understanding the derivative relationship reveals the structure of the motion model
- Two limits needed for the derivation
- When x is measured in radians and approaches zero, sin x / x approaches 1
- (1 − cos x) / x approaches 0
- The derivatives of sine and cosine
- The definition of a derivative and the angle-addition formulas give d(sin x)/dx = cos x
- The same method gives d(cos x)/dx = −sin x
- The repeated-derivative cycle
- Two derivatives produce the negative of the original function
- Four derivatives return to the original function
Introduction
I previously introduced differentiation as a way to find a rate of change—or, from a geometric perspective, the slope of a tangent line. Differentiating position with respect to time gives velocity as a function of time, and differentiating velocity with respect to time gives acceleration as a function of time. (That is starting to sound like a tongue twister.)
In the opening part of high-school kinematics, polynomial functions are often used to explain how position, velocity, and acceleration depend on time.
This time, however, I want to look at the derivatives of two trigonometric functions that matter greatly in physics: the sine function 
and the cosine function 
.
Why do we need the derivatives of sine and cosine? Are derivatives of polynomial functions not enough?
The reason is simple harmonic motion (S.H.M.), a dedicated topic in upper-secondary physics. Its position, velocity, and acceleration as functions of time are frequently expressed with trigonometric functions.
Conversely, integration can also be used to recover the relationships among these functions.
For the basic ideas of differentiation and integration, see “A Gentle Introduction to Differentiation” and “A Gentle Introduction to Integration.”
When simple harmonic motion is introduced, limited class time often means the lesson simply states that the derivative of sine is cosine and the derivative of cosine is negative sine, then moves on.
After thinking it over, I felt that readers who want to know where these formulas come from deserved a short article devoted to the derivation.
Two Important Limits
Let us begin with a diagram commonly used to explore relationships among trigonometric functions:
Before differentiating trigonometric functions, we need a few important limits.
First, consider a limit that is fundamental in both mathematics and physics:
To evaluate it, we will examine the right-hand and left-hand limits separately.
Start with the right-hand limit. Consider the unit circle shown above. Suppose 
makes an angle of x radians with the positive x-axis, where x > 0, and intersects the unit circle at C. The perpendicular feet from A and C to the positive x-axis are B and D, respectively.
We then have the following conditions:
They also satisfy:
Dividing the inequality above by 
gives:
Taking reciprocals gives:
Because 
, the squeeze theorem proves the right-hand limit:
Now consider the left-hand limit. Let 
. From the right-hand limit above, we know that:
Using the properties of trigonometric functions for generalized angles:
Therefore, the left-hand limit is:
The left-hand and right-hand limits both exist and are equal, so we have proved:
Next, a little algebra gives another useful limit:
Begin with the half-angle identity:
A straightforward calculation gives:
Let 
. We then obtain:
Differentiating the Sine Function
Suppose differentiating f(x) with respect to x gives g(x). By definition:
![Equation or diagram: g(x) = d f(x)/dx = limₕ→₀ [f(x+h) − f(x)]/h](../../../pic/work/physicsteaching/physicsteaching11/25.webp)
We call g(x) the derivative of f(x), written as 
.
We now want the derivative of 
, so we need to evaluate the following limit:
![Equation or diagram: limₕ→₀ [sin(x+h) − sin x]/h](../../../pic/work/physicsteaching/physicsteaching11/28.webp)
Using the angle-addition formula:
Substitution gives:
In other words:
Differentiating the Cosine Function
Next, we want the derivative of 
, so we evaluate:
![Equation or diagram: limₕ→₀ [cos(x+h) − cos x]/h](../../../pic/work/physicsteaching/physicsteaching11/33.webp)
Using the angle-addition formula:
Substitution gives:
Therefore:
Putting the Pattern Together
From the proof above and the differentiation formulas, the derivatives of sine follow this pattern:
In other words, differentiating sin x twice gives the negative of the original function, while differentiating it four times completes the cycle.
The derivatives of cosine follow this pattern:
Differentiating cos x twice likewise gives the negative of the original function, and differentiating it four times completes the same cycle.
Conclusion
In high-school trigonometry, we learn that sin x and cos x are closely related through complementary-angle identities and the Pythagorean identity.
This article reveals another connection: differentiation transforms these two functions into one another, while the second derivative of either function is the negative of itself.
Sine and cosine play essential roles in simple harmonic motion. I will return to that application in a future article—stay tuned. :D
FAQ
Simple harmonic motion commonly uses sine and cosine to express position. Differentiating position with respect to time then connects it to velocity and acceleration.
The standard limit sin x / x → 1 as x → 0 holds when x is measured in radians. With degrees, an additional unit-conversion factor appears in the derivative.
Substituting sin x into the definition of a derivative and applying the angle-addition formula and two basic limits gives d(sin x)/dx = cos x.
After expanding cos(x + h), the term involving sin h carries a minus sign, which ultimately gives d(cos x)/dx = −sin x.
Both functions follow a four-step cycle: two derivatives give the negative of the original function, and four derivatives return to the original function.
Share this article:
Recommended Next Step
Trigonometric derivatives connect position, velocity, and acceleration directly in simple harmonic motion. If you want to bring mathematical tools back into high-school physics instead of memorizing formulas in isolation, Wakame’s online physics courses use animation and gradual problem sequences to build the path from concepts to problem solving.
Start getting stronger today!
