Derivatives of Sec, Csc and Cot
The derivatives of secant, cosecant, and cotangent functions can be found using the chain rule and the fundamental trigonometric identities. The derivative of secant is secant multiplied by tangent, the derivative of cosecant is negative cosecant multiplied by cotangent, and the derivative of cotangent is negative cosecant squared. These derivatives are important in calculus and are used in various applications.
4 Key excerpts on "Derivatives of Sec, Csc and Cot"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Notice also that was replaced with 1. This limit was proved in Chapter 2. By the definition of the derivative, By the same process, it could be shown (but will not here) that the derivative of cos(x) is – sin(x). Once these two derivatives are established, the derivatives of the other four basic trigonometric functions can be determined by using the quotient rule. One function, the tangent, is shown below. All six basic trigonometric functions have derivatives, which are listed in Table 4.1. You should commit these derivatives to memory, as they will be used regularly throughout the course. Additionally, you will also need to recall values of the trigonometric functions at key radian measures such as multiples of and You should have learned these in a previous course; review them if you need to. Table 4.1 Function Derivative y = sin(x) = cos(x) y = cos(x) = –sin(x) y = tan(x) = sec 2 (x) y = cot(x) = –csc 2 (x) y = sec(x) = sec(x) · tan(x) y = csc(x) = – csc(x) · cot(x) Close inspection shows that there are patterns in the derivatives that make them easier to learn. Notice that the derivative of each function beginning with “co” has a negative sign. Also notice that the derivatives of each pair of cofunctions are themselves cofunctions. For example, the derivative of tangent involves the secant function, and the derivative of cotangent involves the cosecant function. Watch for a couple of subtleties when working with and evaluating derivatives of trigonometric functions. The first regards the domain of each function. Only sine and cosine are defined for all real numbers; there are values of x for which the derivatives of the other four functions are not defined...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The reciprocal of sine is c o s e c ant (csc), the reciprocal of cosine is sec ant (sec), and the reciprocal of tangent is cot angent (cot): The definitions of the six trigonometric functions are summarized in Figure 9.3. FIGURE 9.3 The six trigonometric functions of acute angle θ To evaluate sec x, csc x, or cot x, determine the value of its reciprocal function. Then find the reciprocal of that value by pressing the calculator’s reciprocal key,. EXAMPLES To find sec 60°, press Thus, sec 60° = 2. To find csc 35°20′ correct to four decimal places, press Thus, csc 35°20′ ≈ 1.7291. COFUNCTION RELATIONSHIPS The prefix “co” in co sine, co secant, and co tangent represents co mplementary. Two angles are complementary when their measures add up to 90°. Pairs of cofunctions have equal values when their angles are complementary: • sin θ ° = cos (90 – θ)° Example: sin 50° = cos 40° • sec θ ° = csc (90 – θ)° Example: sec 24° = csc 66° • tan θ ° = cot (90 – θ)° Example: tan 20°50′ = cot 69° 10′ Lesson 9-3: The General Angle KEY IDEAS By defining trigonometric functions of angles of rotation using coordinates, the domains of the six trigonometric functions can be expanded to include sets of real numbers. As a result, we no longer are limited to positive acute angles. It will now be possible to consider trigonometric functions of angles greater than 90° or less than 0°. STANDARD POSITION FOR A GENERAL ANGLE An angle is in standard position when its vertex is at the origin and one of its sides, called the initial side, remains fixed on the positive x- axis...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...2.10 c. Note that tangent is still a periodic function, but of smaller period, π rad, according to (2.300) whereas combination of Eqs. (2.295), (2.296), and (2.299) implies that (2.301) – so the (trigonometric) tangent is an odd function. The tangent is also a monotonically increasing function – yet it exhibits vertical asymptotes at x = kπ /2 (with relative integer k), see again Fig. 2.10 c. The cotangent of angle θ may, in turn, be defined as the ratio of the length of the adjacent leg, [ OA ], to the length of the opposite leg, [ AB ], in triangle [ OAB ] – or, instead, as the tangent of the complementary of angle θ, i.e. ∠ BOE, via the ratio of the length of the opposite leg, [ BE ], to the length of the adjacent leg, [ OB ], in triangle [ OBE ], viz. (2.302) as outlined in Fig. 2.10 a, where Eq. (2.287) was taken advantage of; Eq. (2.302) may be redone to (2.303) again after dividing numerator and denominator by, and recalling Eqs. (2.288) and (2.290). For a general argument x (in rad), one may accordingly state (2.304) following comparative inspection of Eqs. (2.298) and (2.303) – which varies with argument x as depicted in Fig. 2.10 d. Once again, a period of π rad is apparent, i.e. (2.305) while Eqs. (2.301) and (2.304) imply (2.306) – meaning that cotangent is also an odd function. The cotangent always decreases when x increases, and is driven by vertical asymptotes described by x = kπ (with relative integer k) as can be perceived in Fig. 2.10 d. With regard to secant of angle θ, it follows from the ratio of the length of the hypotenuse, [ OB ], to the length of the adjacent leg, [ OA ], in triangle [ OAB ] – or, alternatively, as the ratio of the length of the hypotenuse, [ OD ], to the length of the adjacent leg, [ OB ], in triangle [ OBD ], according to (2.307) – as outlined in Fig. 2.10 a, also at the expense of Eq. (2.287) ; one may rewrite Eq. (2.307) as (2.308) after taking the reciprocal of the reciprocal, in view of Eq. (2.288)...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...For instance, given the table of values of f (x) below, and given that f (x) is continuous and differentiable, approximate f ′(3). You will not be told to use the slope of the secant between two points containing x = 3, you’ll just have to know to do this. So, or. There can be different answers since this is only an approximation. X f (x) 2 1.3 3 1.6 5 2.9 6 2.8 B. Notation of Derivative and common terms used to describe it 1. Common notations: 2. Common terms to describe the derivative: instantaneous rate of change, change in y with respect to x, slope. C. Definition of Derivative 1. Derivative as a function: 2. Derivative at a point, (Notice that this is equivalent to. This is to say that the slope of the tangent line at x = a is equal to the limit of the slope of the secant line between x = a (and any other x -value as the x -value approaches a.). D. Existence of Derivative at a point A function’s derivative does not exist at points where the function has a discontinuity, corner, cusp, vertical asymptote or vertical. tangent. 1. y = f (x) f ′(0) does not exist because f (x) is discontinuous at x = 0. 2. y = g (x) g ′(1) does not exist because g (x) has a corner at x = 1 and the left and right derivatives are not equal. 3. y = h (x) h ′(0) does not exist because at x = 0 h (x) has a cusp (also a vertical tangent) 4. y = s (x) s ′(0) does not exist because s (x) has a vertical tangent at x = 0 E. Properties. of f (x) = e x and g (x) = ln(x) 1. e ln(x) = x 2. ln(e x) = x 3. ln(xy) = ln(x) + ln(y) 5. ln(x y) = y ln(x) 6. ln(1) = 0 7. ln(0) does not exist F. L’Hôspital’s Rule—allows you to take limits that have indeterminate forms, such as or. So, if equals one of these indeterminate forms, then take. Note that you are not using the quotient rule here, you are simply taking the derivative of the numerator and denominator separately. If the limit still has an indeterminate form then repeat the process as necessary...
