19.Derivative of sine and cosine JJ II

19. Derivative of sine and cosine

19.1. Two trigonometric limits

The rules for finding the derivative of the functions sin x and cos x depend on two limits (that are used elsewhere in calculus as well):

Two trigonometric limits.

sin

(a) lim

= 1,

0

cos - 1

(b) lim

= 0.

0

The verification we give of the first formula is based on the pictured wedge of the unit circle:

Derivative of sine and cosine Two trigonometric limits Statement Examples

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Derivative of sine and cosine

Two trigonometric limits Statement Examples

The segment tagged with sin has this length by the definition of sin as the y-coordinate of the point P on the unit circle corresponding to the angle . The line segment tagged with tan has this length since looking at the large triangle, we have tan = o/a = o. The arc tagged with has this length by the definition of radian measure of an angle. The diagram reveals the inequalities

sin < < tan .

The first inequality implies (sin )/ < 1; the second says < (sin )/(cos ), implying that

cos < (sin )/. Therefore,

sin

cos <

< 1.

As goes to 0, both ends go to 1 forcing the middle expression to go to 1 as well (by the squeeze theorem). This establishes (a).

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For the second formula, we use a method that is similar to our rationalization method, as well as the main trigonometric identity, and finally the first formula:

cos - 1

cos - 1 cos + 1

lim

= lim

?

0

0

cos + 1

cos2 - 1 = lim

0 (cos + 1)

- sin2 = lim

0 (cos + 1)

sin sin

= - lim

?

0 cos + 1

sin

sin

= - lim

? lim

0 0 cos + 1

= -1 ? 0 = 0.

This completes the verification of the two trigonometric limit formulas.

19.2. Statement

Derivative of sine and cosine. d

(a) [sin x] = cos x, dx d

(b) [cos x] = - sin x. dx

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We verify only the first of these derivative formulas. With f (x) = sin x, the formula says f (x) = cos x:

f (x + h) - f (x)

f (x) = lim

h0

h

sin(x + h) - sin x

= lim

h0

h

(sin x cos h + cos x sin h) - sin x

= lim

h0

h

cos h - 1

sin h

= lim sin x ?

+ cos x ?

h0

h

h

cos h - 1

sin h

= sin x ? lim

+ cos x ? lim

h0 h

h0 h

= sin x ? 0 + cos x ? 1

= cos x.

((4) of 4.3) ((b) and (a))

The formula says that f (x) = sin x has general slope function f (x) = cos x, so the height of the graph of the cosine function at x should be the slope of the graph of the sine function at x. The following figures show this relationship for x a multiple of /2.

Derivative of sine and cosine Two trigonometric limits Statement Examples

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Derivative of sine and cosine

Two trigonometric limits Statement Examples

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19.3. Examples

19.3.1 Example Find the derivative of f (x) = 3 cos x + 5 sin x. Solution We use the rules of this section after first applying the sum rule and the constant

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