3.4. Derivatives of Trigonometric Functions. Derivative of the Sine Function

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1 3.4 Derivatives of Trigonometric Functions Derivatives of Trigonometric Functions Many of the phenomena we want information about are approximately perioic (electromagnetic fiels, heart rhythms, ties, weather). The erivatives of sines an cosines play a key role in escribing perioic changes. This section shows how to ifferentiate the six basic trigonometric functions. Derivative of the Sine Function To calculate the erivative of ƒsx sin x, for x measure in raians, we combine the its in Example 5a an Theorem 7 in Section 2.4 with the angle sum ientity for the sine: sin sx + h sin x cos h + cos x sin h.

2 84 Chapter 3: Differentiation If ƒsx sin x, then ƒsx + h - ƒsx ƒ sx sin sx + h - sin x ssin x cos h + cos x sin h - sin x sin x scos h - + cos x sin h asin x # cos h - b + acos x # sin h b sin x # cos h - sin x # + cos x # cos x. + cos x # sin h Derivative efinition Sine angle sum ientity Example 5(a) an Theorem 7, Section 2.4 The erivative of the sine function is the cosine function: ssin x cos x. x EXAMPLE Derivatives Involving the Sine (a) (b) (c) y x 2 - sin x: y x 2 sin x: y sin x x : y x 2x - Asin xb x 2x - cos x. y x x2 Asin xb + 2x sin x x x 2 cos x + 2x sin x. y x # x x Asin xb - sin x # x 2 x cos x - sin x x 2. Difference Rule Prouct Rule Quotient Rule Derivative of the Cosine Function With the help of the angle sum formula for the cosine, cos sx + h cos x cos h - sin x sin h,

3 3.4 Derivatives of Trigonometric Functions 85 y y' y cos x x y' sin x FIGURE 3.23 The curve y -sin x as the graph of the slopes of the tangents to the curve y cos x. x we have cossx + h - cos x scos x x scos x cos h - sin x sin h - cos x cos xscos h - - sin x sin h cos x # cos h - cos x # cos h - cos x # - sin x # -sin x. - sin x # sin h - sin x # sin h Derivative efinition Cosine angle sum ientity Example 5(a) an Theorem 7, Section 2.4 The erivative of the cosine function is the negative of the sine function: scos x -sin x x Figure 3.23 shows a way to visualize this result. EXAMPLE 2 Derivatives Involving the Cosine (a) (b) (c) y 5x + cos x: y sin x cos x: y cos x - sin x : y A - sin xb x - sin x s - sin x 2 s - sin xs -sin x - cos xs - cos x - sin x. y x x s5x + Acos xb x 5 - sin x. y x sin x x Acos xb + cos x Asin xb x sin xs -sin x + cos xscos x cos 2 x - sin 2 x. x Acos xb - cos x A - sin xb x s - sin x 2 s - sin x 2 Sum Rule Prouct Rule Quotient Rule sin 2 x + cos 2 x

4 86 Chapter 3: Differentiation Simple Harmonic Motion 5 The motion of a boy bobbing freely up an own on the en of a spring or bungee cor is an example of simple harmonic motion. The next example escribes a case in which there are no opposing forces such as friction or buoyancy to slow the motion own. s 5 Rest position Position at t EXAMPLE 3 Motion on a Spring A boy hanging from a spring (Figure 3.24) is stretche 5 units beyon its rest position an release at time t to bob up an own. Its position at any later time t is s 5 cos t. What are its velocity an acceleration at time t? FIGURE 3.24 A boy hanging from avertical spring an then isplace oscillates above an below its rest position. Its motion is escribe by trigonometric functions (Example 3). s, y 2 y 5 sin t s 5 cos t 5 2 FIGURE 3.25 The graphs of the position an velocity of the boy in Example 3. t Solution We have Position: s 5 cos t Velocity: y s t s5 cos t -5 sin t t Acceleration: a y s -5 sin t -5 cos t. t t Notice how much we can learn from these equations:. As time passes, the weight moves own an up between s -5 an s 5 on the s-axis. The amplitue of the motion is 5. The perio of the motion is 2p. 2. The velocity y -5 sin t attains its greatest magnitue, 5, when cos t, as the graphs show in Figure Hence, the spee of the weight, ƒ y ƒ 5 ƒ sin t ƒ, is greatest when cos t, that is, when s (the rest position). The spee of the weight is zero when sin t. This occurs when s 5 cos t ;5, at the enpoints of the interval of motion. 3. The acceleration value is always the exact opposite of the position value. When the weight is above the rest position, gravity is pulling it back own; when the weight is below the rest position, the spring is pulling it back up. 4. The acceleration, a -5 cos t, is zero only at the rest position, where cos t an the force of gravity an the force from the spring offset each other. When the weight is anywhere else, the two forces are unequal an acceleration is nonzero. The acceleration is greatest in magnitue at the points farthest from the rest position, where cos t ;. EXAMPLE 4 Jerk The jerk of the simple harmonic motion in Example 3 is j a t s -5 cos t 5 sin t. t It has its greatest magnitue when sin t ;, not at the extremes of the isplacement but at the rest position, where the acceleration changes irection an sign. Derivatives of the Other Basic Trigonometric Functions Because sin x an cos x are ifferentiable functions of x, the relate functions tan x cos sin x cos x x, cot x sin x, sec x cos x, an csc x sin x

5 3.4 Derivatives of Trigonometric Functions 87 are ifferentiable at every value of x at which they are efine. Their erivatives, calculate from the Quotient Rule, are given by the following formulas. Notice the negative signs in the erivative formulas for the cofunctions. Derivatives of the Other Trigonometric Functions x stan x sec2 x ssec x sec x tan x x x scot x -csc2 x scsc x -csc x cot x x To show a typical calculation, we erive the erivative of the tangent function. The other erivations are left to Exercise 5. EXAMPLE 5 Fin (tan x) > x. Solution x Atan xb x a cos sin x cos x x b cos x cos x - sin x s -sin x cos 2 x cos2 x + sin 2 x cos 2 x cos 2 x sec2 x x Asin xb - sin x Acos xb x cos 2 x Quotient Rule EXAMPLE 6 Fin y if y sec x. Solution y sec x y sec x tan x y ssec x tan x x sec x x Atan xb + tan x Asec xb x Prouct Rule sec xssec 2 x + tan xssec x tan x sec 3 x + sec x tan 2 x

6 88 Chapter 3: Differentiation The ifferentiability of the trigonometric functions throughout their omains gives another proof of their continuity at every point in their omains (Theorem, Section 3.). So we can calculate its of algebraic combinations an composites of trigonometric functions by irect substitution. EXAMPLE 7 Fining a Trigonometric Limit x: 22 + sec x cossp - tan x 22 + sec cossp - tan 22 + cossp