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19 19.1 Derivative of sine and cosine 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 θ = 1, θ→0 θ (a) lim (b) lim θ→0 cos θ − 1 = 0. θ The verification we give of the first formula is based on the pictured wedge of the unit circle: The segment tagged with sin θ has this length by the definition of sin θ as the y-coordinate of the point P on the unit circle corresponding to the angle θ. The line segment tagged with tan θ has this length since looking at the large triangle, we have tan θ = o/a = o. The arc tagged with θ has this length by the definition of radian measure of an angle. The diagram reveals the inequalities sin θ < θ < tan θ. The first inequality implies (sin θ)/θ < 1; the second says θ < (sin θ)/(cos θ), implying that cos θ < (sin θ)/θ. Therefore, cos θ < sin θ < 1. θ As θ 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). 1 19 DERIVATIVE OF SINE AND COSINE 2 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: lim θ→0 cos θ − 1 cos θ − 1 cos θ + 1 = lim · θ→0 θ θ cos θ + 1 cos2 θ − 1 = lim θ→0 θ(cos θ + 1) − sin2 θ θ→0 θ(cos θ + 1) sin θ sin θ = − lim · θ→0 θ cos θ + 1 sin θ sin θ = − lim · lim θ→0 θ θ→0 cos θ + 1 = −1 · 0 = 0. = lim This completes the verification of the two trigonometric limit formulas. 19.2 Statement Derivative of sine and cosine. (a) d [sin x] = cos x, dx (b) d [cos x] = − sin x. dx We verify only the first of these derivative formulas. With f (x) = sin x, the formula says f 0 (x) = cos x: f (x + h) − f (x) h→0 h sin(x + h) − sin x = lim h→0 h (sin x cos h + cos x sin h) − sin x = lim h→0 h cos h − 1 sin h = lim sin x · + cos x · h→0 h h cos h − 1 sin h = sin x · lim + cos x · lim h→0 h→0 h h = sin x · 0 + cos x · 1 f 0 (x) = lim = cos x. ((4) of 4.3) ((b) and (a)) 19 DERIVATIVE OF SINE AND COSINE 3 The formula says that f (x) = sin x has general slope function f 0 (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 π/2. 19.3 19.3.1 Examples 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 multiple rule: d [3 cos x + 5 sin x] dx d d = 3 [cos x] + 5 [sin x] dx dx = 3(− sin x) + 5(cos x) f 0 (x) = = −3 sin x + 5 cos x. 19.3.2 Example Find all points on the graph of f (x) = sin x at which the tangent line has slope 1/2. Solution The general slope function for this function is its derivative, which is f 0 (x) = cos x. We get the x-coordinates of the desired points by solving 19 DERIVATIVE OF SINE AND COSINE 4 f 0 (x) = 1/2, that is, cos x = 1/2. Picturing the unit circle and using the 30-6090 triangle we find that x = π/3 (60◦ ) in the first quadrant yields a cosine of 1/2, as does x = −π/3 in the fourth quadrant. A multiple of 2π added to these angles produces the same cosine. Therefore, the x-coordinates of the desired points are π/3 + 2πn, −π/3 + 2πn (n any integer). √ The y-coordinate corresponding to x = π/3 is f (π/3) = sin(π/3) = 3/2 and this is also the y-coordinate corresponding to x = π/3+2πn √ for any n. Similarly, the y-coordinate corresponding to x = −π/3+2πn is − 3/2. The desired points are √ √ (n any integer). (π/3 + 2πn, 3/2), (−π/3 + 2πn, − 3/2) 19 – Exercises 19 – 1 Find all points on the graph of f (x) = cos x at which the tangent line has slope √ 3/2.