Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
3.3: Derivatives of Trigonometric Functions Telyn Kusalik 3.3.1 Derivatives of sin and cos Objective: Find the derivative of f (x) = 2 sin x + 3 cos x. MATH 111 Fall 2015 Derivatives of sin and cos graphically Let’s look at the graphs of y = sin x and y = cos x and attempt to take the derivatives of these functions graphically. The graph of the derivative of y = sin x looks an awful lot like the graph of y = cos x. The graph of the derivative of y = cos x looks like an upside-down sin x. These are not coincidences. The derivatives of sin x and cos x are as follows. (sin x)0 = cos x (cos x)0 = − sin x Radian Measure and Calculus The fact that the derivative of sin x is cos x is a fact that crucially depends on us using radian measure. If the sin function is graphed using radian measure, the slope when x = 0 is exactly 1, and cos(0) = 1. However, if we graph sin x using degree measure, we notice that the slope at x = 0 is a lot smaller than 1. It turns out to be π exactly 180 . Using degree measure, the rule for the derivative of sin x would be: π (sin x)0 = cos x 180 So the reason we use radian measure is that it makes the calculus of trigonometric functions much simpler. Derivations of Derivative Rules for sin and cos Derivatives Involving sin x and cos x In order to derive the derivative rules for sin and cos, we need the two limits you found on your first assignment: lim x→0 sin x =1 x lim x→0 cos x − 1 =0 x We also need the sum identities for sin and cos: sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b − sin a sin b Asking you to find the derivative of sin x or cos x on its own is too easy a problem for me to ask. Usually, I’ll ask problems involving the product or quotient rule as well as a sin or cos: Example 1: Find the derivative of f (x) = Example 2: Differentiate y = √ x sin x. cos x 1−sin x Using these tools, we can use the definition of the derivative to find the derivatives of sin and cos. I will derive the derivative d formula dx (sin x) = cos x as an example. Higher Derivatives of sin and cos Tangent Line Problems One of the interesting things about sin and cos functions is that when we take higher and higher derivatives of them, we end up getting back where we started. Let’s start with y = cos x y 0 = − sin x y 00 = − cos x y 000 = sin x y (4) = cos x y (5) = − sin x y (6) = − cos x y (7) = sin x y (8) = cos x In general, any derivative of order which is a multiple of 4 will be cos x. This makes it easy to find very high derivatives like: d 43 cos x. dx 43 The 40th derivative is cos x, so the 41st is − sin x, so the 42nd is − cos x, so the 43rd is sin x. Just like with other types of functions, tangent line problems can be written than involve trigonometric functions. The difficulty many students have is not with the actual calculus of these functions, but in applying pre-calculus skills such as solving trigonometric equations. Example 3 : For what values of x does the graph of f have a horizontal tangent? f (x) = ex cos x Derivatives of tan, cot, sec and csc using sin and cos 3.3.2 Derivatives of other Trigonometric Functions Objective: Find the derivative of f (x) = 2 sec x + 3 tan x. Since the trigonometric functions tan, cot, sec, and csc can all be expressed using sin and cos, their derivatives can be taken using the quotient rule: 0 (tan x) = sin x cos x 0 = cos2 x + sin2 x = sec2 x cos2 x I could ask you, on a quiz or test, to “prove” the derivative rule for one of these trigonometric functions: d Example 4: Prove that dx (sec x) = sec x tan x. How to Remember Trigonometric Derivatives Most students usually choose to memorize the derivatives of tan, cot, sec and csc rather than to go through the derivations like on the previous slide. One thing to keep in mind to help remember them is that the derivatives of the “co” functions are basically the same as the derivatives of the “original” functions except they are “co” and negative: 0 0 (sin x) = cos x (cos x) = − sin x (tan x)0 = sec2 x (cot x)0 = − csc2 x (sec x)0 = sec x tan x (csc x)0 = − csc x cot x Derivatives Involving tan, cot, sec and csc Again, just like with sin and cos, most derivative problems in this module will involve the product and quotient rule as well as the trigonometric functions. Example 5: Differentiate g(θ) = eθ (tan θ − θ). Example 6: Find the derivative of y = 1−sec x tan x . More Complicated Derivatives The most complicated derivatives I can ask for at this point are those that involve more than one product or quotient rule: Example 7: Differentiate y = x 2 sin x tan x.