3.5 Derivatives of Trigonometric Functions

One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.

Derivatives of the Sine and Cosine Functions

We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function $f (x),$

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h} .

Consequently, for values of $h$ very close to 0, $f^{'} (x) \approx \frac{f ( x + h ) - f ( x )}{h} .$ We see that by using $h = 0.01,$

\frac{d}{d x} (sin x) \approx \frac{sin ( x + 0.01 ) - sin x}{0.01}

By setting $D (x) = \frac{s i n ( x + 0.01 ) - s i n x}{0.01}$ and using a graphing utility, we can get a graph of an approximation to the derivative of $sin x$ (Figure 3.25).

The function D(x) = (sin(x + 0.01) − sin x)/0.01 is graphed. It looks a lot like a cosine curve. — Figure 3.25The graph of the function $D (x)$ looks a lot like a cosine curve.

Upon inspection, the graph of $D (x)$ appears to be very close to the graph of the cosine function. Indeed, we will show that

\frac{d}{d x} (sin x) = cos x .

If we were to follow the same steps to approximate the derivative of the cosine function, we would find that

\frac{d}{d x} (cos x) = - sin x .

Proof

Because the proofs for $\frac{d}{d x} (sin x) = cos x$ and $\frac{d}{d x} (cos x) = - sin x$ use similar techniques, we provide only the proof for Before beginning, recall two important trigonometric limits we learned in Introduction to Limits:

h \to 0 lim \frac{sin h}{h} = 1 and h \to 0 lim \frac{cos h - 1}{h} = 0.

The graphs of $y = \frac{( s i n h )}{h}$ and $y = \frac{( c o s h - 1 )}{h}$ are shown in Figure 3.26.

The function y = (sin h)/h and y = (cos h – 1)/h are graphed. They both have discontinuities on the y-axis. — Figure 3.26These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.

We also recall the following trigonometric identity for the sine of the sum of two angles:

sin (x + h) = sin x cos h + cos x sin h .

Now that we have gathered all the necessary equations and identities, we proceed with the proof.

\frac{d}{d x} sin x = lim_{h \to 0} \frac{sin ( x + h ) - sin x}{h} = lim_{h \to 0} \frac{sin x cos h + cos x sin h - sin x}{h} = lim_{h \to 0} (\frac{sin x cos h - sin x}{h} + \frac{cos x sin h}{h}) = lim_{h \to 0} (sin x (\frac{cos h - 1}{h}) + cos x (\frac{sin h}{h})) = sin x \cdot 0 + cos x \cdot 1 = cos x Apply the definition of the derivative. Use trig identity for the sine of the sum of two angles. Regroup. Factor out sin x and cos x . Apply trig limit formulas. Simplify.

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Figure 3.27 shows the relationship between the graph of $f (x) = sin x$ and its derivative $f^{'} (x) = cos x .$ Notice that at the points where $f (x) = sin x$ has a horizontal tangent, its derivative $f^{'} (x) = cos x$ takes on the value zero. We also see that where is increasing, and where is decreasing,

The functions f(x) = sin x and f’(x) = cos x are graphed. It is apparent that when f(x) has a maximum or a minimum that f’(x) = 0. — Figure 3.27Where $f (x)$ has a maximum or a minimum, $f^{'} (x) = 0$ that is, $f^{'} (x) = 0$ where $f (x)$ has a horizontal tangent. These points are noted with dots on the graphs.

Derivatives of Other Trigonometric Functions

Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.

Higher-Order Derivatives

The higher-order derivatives of $sin x$ and $cos x$ follow a repeating pattern. By following the pattern, we can find any higher-order derivative of $sin x$ and $cos x .$

Key Concepts

We can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. The results are
$\frac{d}{d x} sin x = cos x \frac{d}{d x} cos x = - sin x .$

Key Equations

Derivative of sine function	$\frac{d}{d x} (sin x) = cos x$
Derivative of cosine function	$\frac{d}{d x} (cos x) = - sin x$