Now that we’ve learned how to differentiate functions in which the variable is raised to a power, let’s look at two functions that work in a different way: the sine and the cosine.
What’s the sine?
In trigonometry, you learned that the sine function acts on angles. The unit-circle model (Fig. 7-1) defines the sine of x (sin x ) for all possible angles x. The value of x is given in radians (rad).
Positive angles go counterclockwise from the right-hand horizontal axis or “due east.” Nega-tive angles go clockwise from “due east.” In Fig. 7-1, each axis division represents 1/4 unit, so the circle has a radius of 1 unit.
The sine of an angle is the vertical-axis coordinate of the point where the radial ray inter-sects the unit circle. Because this circle has a radius of 1 unit, the sine function can never attain values larger than 1 or smaller than −1. Figure 7-1 shows the situation for an angle of 2p/3 rad (x= 2p/3), which is 1/3 of a rotation.
Remember that all angle expressions for trigonometric functions are given in radians (never in degrees) unless you’re told specifically otherwise!
What’s the cosine?
The cosine function can also be defined in terms of the unit circle, as shown in Fig. 7-2. As before, positive angles are expressed as counterclockwise rotation of the radial ray, and nega-tive angles are expressed as clockwise rotation.
The cosine of the angle is the horizontal-axis coordinate of the point where the radial ray intersects the circle. As with the sine, the cosine can never be larger than 1 or smaller than −1.
This drawing illustrates the case where x= 2p/3.
Both the sine and the cosine functions are defined, continuous, and differentiable over the entire set of real numbers. An angle of 2p represents a complete counterclockwise rotation of the ray extending straight out from the origin. Angles larger than 2p represent more than one complete counterclockwise rotation. Negative angles go clockwise instead of counter-clockwise. For example, an angle of −2p/3 translates to 1/3 of a clockwise rotation. An angle of−2p is a full clockwise rotation. Angles smaller than −2p (or larger negatively ) represent more than a full clockwise rotation.
Derivative of the sine
Suppose we graph the function f (x ) = sin x for values of x between −3p and 3p. The result is the solid curve in Fig. 7-3. Because of its wavelike appearance, this curve is often called a sine wave. If we examine it closely, we can see that its slope is 0 when it reaches a crest or local maximum, and the slope is also 0 when it reaches a trough or local minimum. A point of maxi-mum slope occurs 1/4 of a rotation, or p/2 rad, after every trough. A point of minimaxi-mum slope
Sine and Cosine Functions 109
sinx
Figure 7-1 The unit-circle model for the sine function. The angle x is given in radians, and is measured from the right-hand horizontal axis or “due east.” Each axis division represents 1/4 unit.
cosx
x
cosine Ray from
origin
Each axis increment is 1/4 unit
Circle with radius of 1 unit
Figure 7-2 The unit-circle model for the cosine function.
Each axis division represents 1/4 unit.
–3p
3p 3
2
1
–1
–2
–3
x Value of
function
f(x) = sin x f¢(x) = cos x
Figure 7-3 The derivative of the sine is the cosine.
This can be seen when the two functions are graphed as waves and then compared.
occursp /2 rad after every crest. The curve crosses the x axis at every point where the slope is maximum or minimum.
If we plot a graph of the slope of the sine function, which represents its derivative, for many points along the x axis, we’ll see that the graph of the derivative is another wave, shown by the dashed curve in Fig. 7-3. It has the same shape and size as the sine wave. This particular wave shape occurs often in physics and engineering. The general term for it is sinusoid.
In Fig. 7-3, we can see that the derivative wave is displaced by p /2 rad to the left of the original wave. This amount of displacement is sometimes called a quarter wavelength or a quarter cycle. In this example, the dashed wave represents the cosine function. Therefore, if we differentiate
f (x ) = sin x we get
f ′(x ) = cos x
Derivative of the cosine
Now let’s graph the function g (x ) = cos x for values of x between −3p and 3p. We get the solid curve shown in Fig. 7-4. This wave, like the sine wave, has a slope of 0 when it reaches a crest or a trough. Also like the sine wave, a point of maximum slope occurs p /2 rad after Sine and Cosine Functions 111
–3p
3p 3
2
1
–1
–2
–3
x Value of
function
g(x) = cos x g¢(x) = –sin x
Figure 7-4 The derivative of the cosine is the negative of the sine.
every trough, and a point of minimum slope occurs p /2 rad after every crest. The cosine wave is a sinusoid, just like the sine wave. The only difference between the cosine wave and the sine wave is the horizontal position, also called the phase.
When we graph the slope of the cosine function for many points, we find that the graph of the derivative is another sinusoid, shown by the dashed curve in Fig. 7-4. The derivative wave is, as in the earlier case with the sine wave, displaced p /2 rad to the left of the original. This new wave is an “upside-down” sine wave, so it represents the negative of the sine function. If
g (x)= cos x then the derivative is
g ′(x ) = − sinx
Example
Now that we’ve learned how to differentiate two new functions, we can differentiate any func-tion that contains them in a sum, product, reciprocal, or quotient. We can also use the chain rule to differentiate composite functions containing them. Let’s try an example:
f (x ) = −3x4+ 2 cos x Using the sum rule, we know that
f ′(x ) =d /dx (−3x4)+d / dx (2 cos x )
The power rule and the multiplication-by-constant rule allow us to rewrite this as f ′(x ) = −12x 3+ 2 d /dx (cos x )
We know that the derivative of the cosine is the negative of the sine. Therefore f ′(x ) = −12x 3+ 2 · (− sinx ) = −12x 3− 2 sin x
Are you confused?
The little dot in the first part of the above equation represents multiplication. It’s often used instead of the traditional “times sign,” which some people think looks like the letter x representing a variable. If we were to write the first part of the above equation with the conventional “times sign,” we would get
f ′(x ) = −12x3+ 2 × (− sinx ) Some people might confuse this with the completely different equation
f ′(x ) = −12x 3+ 2x (− sinx )
Let’s agree that from now on, whenever we feel tempted to use a “times sign” to represent multiplication, we’ll use the dot. This dot should be elevated above the base line. Otherwise, it could be confused with a decimal point! But let’s also remember that we can often indicate multiplication without writing any symbol at all, as in expressions such as 2x, xy, or 12ab2x 3.
Another example
Let’s try differentiating a product of trigonometric functions. Consider q (x ) = sin x cos x
We can name the individual functions as
f (x ) = sin x and
g (x ) = cos x The derivatives are
f ′(x ) = cos x and
g ′(x ) = − sinx Applying the product rule for differentiation gives us
q ′(x ) = (f g)′ (x ) = f ′(x ) g (x ) +g ′(x ) f (x )
= cos x cos x+ (− sinx ) sin x = (cosx )2− (sin x )2
With positive integer powers of trigonometric functions such as the sine and the cosine, the exponent is customarily written directly after the abbreviated name of the function.
For example, instead of (sin x )2, we write sin2 x. We can therefore rewrite the above equation as
q ′(x ) = cos 2x− sin2x
Be warned!
The above notational trick is never used for negative integer powers of trig functions. For example, when we write sin−1x, we actually mean the inverse of the sine of x, also called the Arcsine. This isn’t even a function unless its domain is restricted.
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Here’s a challenge!
Consider a function p (x ) that we get when we take sin x and then square the result:
p (x ) = sin2x
Differentiate this function. Indicate the values, if any, for which it is nondifferentiable.
Solution
We must use the chain rule in this situation. Let’s name the component functions like this:
f (x ) = sin x
and
g ( y ) =y2
The derivatives of the component functions are
f ′(x ) = cos x
and
g ′( y ) = 2y
Now we can write the derivative of the composite function using the chain rule:
p′(x ) = {g [ f (x )]}′ = g ′ [ f (x )] · f ′(x ) = 2 sin x cos x
Both of the component functions are continuous and differentiable over the entire set of reals. We never get a denominator that can become equal to 0, because we never have to divide by anything. Therefore, the composite function is differentiable over the entire set of reals.