Why Derive the Sine Derivative?
A derivative describes the instantaneous rate of change of a function. For the sine function, the derivative turns out to be the cosine function. Rather than accept that formula as a fact to memorize, we can obtain it directly from the definition of a derivative.
The derivation depends on a precise trigonometric identity and two limits. It also depends on measuring angles in radians. Each part has a distinct role: the identity rewrites the difference quotient, while the limits evaluate the pieces that remain as the input increment approaches zero.
The Difference Quotient for Sine
For a function \(f\), its derivative at \(x\), when the limit exists, is defined by the difference quotient
Here \(h\) is a nonzero increment that approaches zero. For \(f(x)=\sin x\), this definition gives
The quotient compares the sine value at \(x+h\) with the sine value at \(x\), then divides their difference by the change in input. To evaluate its limit, use the angle-addition identity
The standard trigonometric limits needed are
These limits apply when the angle variable is measured in radians. In particular, the first limit has value \(1\) under the radian convention used here.
Deriving the Formula
Substitute the angle-addition identity into the difference quotient. Then regroup the numerator so that each standard limit appears explicitly.
In this limit, \(x\) is fixed, so \(\sin x\) and \(\cos x\) are constants with respect to \(h\). Taking the limit term by term gives
Why the Derivation Works
The angle-addition identity makes the change in sine depend on two separate changes: the factor \(\cos h-1\) and the factor \(\sin h\). After division by \(h\), the first contributes a limit of zero, while the second contributes a limit of one. The remaining coefficients are \(\sin x\) and \(\cos x\), so only the cosine term survives.
This also establishes that the derivative exists at every real input. The two limits used in the calculation exist, and the factors \(\sin x\) and \(\cos x\) are finite constants for each fixed real \(x\). Therefore, the difference quotient has a finite limit for each such \(x\).
Write the difference quotient for \(\sin x\).
Apply the angle-addition identity to \(\sin(x+h)\).
Factor the quotient into terms involving \((\cos h-1)/h\) and \(\sin h/h\).
The two limits are zero and one, respectively, leaving \(\cos x\).
Worked Examples
Worked Example: Find the Slope of Sine at Zero from the Definition
At \(x=0\), the difference quotient is
Since \(\sin 0=0\), this becomes
Thus the slope of the sine function at zero is \(1\). The general formula gives the same value: \(\cos 0=1\). This checks the result at this particular input.
Worked Example: Find the Slope of Sine at \(\pi/3\)
Use the difference quotient directly at \(x=\pi/3\):
Apply the angle-addition identity and use \(\sin(\pi/3)=\sqrt{3}/2\) and \(\cos(\pi/3)=1/2\):
Now take the limit of each term:
Therefore, the slope is \(1/2\). The formula gives \(\cos(\pi/3)=1/2\), which verifies the calculation.
Worked Example: Write the Tangent Line to Sine at \(\pi/6\)
The point on the graph when \(x=\pi/6\) has height
The derivative formula gives the slope at that input:
A line with slope \(m\) through \((a,b)\) has equation \(y-b=m(x-a)\). Substituting \(a=\pi/6\), \(b=1/2\), and \(m=\sqrt{3}/2\) gives
Check: when \(x=\pi/6\), the right side is zero, so \(y=1/2=\sin(\pi/6)\). The line passes through the correct point, and its slope is \(\sqrt{3}/2\), the derivative value there.
A Common Mistake
A tempting but incorrect shortcut is to treat the change in sine as if it were simply \(\sin h\). For example, at \(x=\pi/3\), this mistake would replace
with \(\sin h\), leading to the incorrect quotient \(\sin h/h\) and the incorrect slope \(1\). The angle-addition identity shows why that replacement fails:
After division by \(h\), the first term tends to zero and the second tends to \(1/2\), not \(1\). The correction is to expand the shifted sine before taking the limit; the shift changes the sine value according to both terms in the identity.
Why Radians Matter Here
The standard limits used in the proof are stated for \(h\) in radians. The derivation relies on their exact values: \(\sin h/h\) approaches \(1\), and \((\cos h-1)/h\) approaches \(0\). If the angle variable were measured in another unit, the first limit would not have the same value, and the resulting derivative formula would include a conversion factor.
For this reason, the result proved here is specifically the derivative of sine when its input is in radians. The unit is part of the mathematical condition, not an optional label.
Using the Result
The formula translates a derivative question into a cosine value. For instance, to find the instantaneous rate of change of \(\sin x\) at any specified radian input \(a\), evaluate \(\cos a\). The difference-quotient derivation explains why this rule holds for every real \(a\), rather than merely confirming it at a few sample values.
In applications, this derivative can describe the rate at which an oscillating quantity changes at a particular position in its cycle. The derivative may be positive, negative, or zero depending on the cosine value. The sign indicates whether sine is increasing, decreasing, or momentarily has a horizontal tangent at that input.
Check Your Understanding
Answer these questions using the difference quotient, the two standard limits, and the derivative formula established above.
- Write the difference quotient for the derivative of \(\sin x\) at a real input \(x\).
- Which angle-addition identity is used to expand \(\sin(x+h)\)?
- Evaluate the derivative of sine at \(x=\pi/4\).
- Why does the term involving \((\cos h-1)/h\) contribute zero in the derivation?
- What incorrect conclusion results from replacing \(\sin(x+h)-\sin x\) with \(\sin h\)?
