Derivatives of Sin, Cos and Tan
The derivatives of sin, cos, and tan are fundamental concepts in calculus. The derivative of sin(x) is cos(x), the derivative of cos(x) is -sin(x), and the derivative of tan(x) is sec^2(x). These derivatives are used to find the rate of change of trigonometric functions and are essential in solving various mathematical problems involving trigonometric functions.
6 Key excerpts on "Derivatives of Sin, Cos and Tan"
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...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...2.10 b, as a function of x (expressed in that unit). Note their periodic nature, with period 2 π rad, i.e. (2.292) and (2.293) and also their lower and upper bounds, i.e. − 1 and 1. It becomes apparent from inspection of Fig. 2.10 b that the plot of cos x may be obtained from the plot of sin x via a horizontal translation of π /2 rad leftward; in other words, (2.294) – and such a complementarity to a right angle, of amplitude π /2 rad, justifies the term cosine (with prefix ‐co standing for complementary, or adding up to a right angle). The sine is an odd function, i.e. (2.295) hence, its plot is symmetrical relative to the origin of coordinates. Conversely, the cosine is an even function, i.e. (2.296) – meaning that its plot is symmetrical relative to the vertical axis. The tangent of angle θ may be defined as the ratio of the length of the opposite leg, [ AB ], to the length of the adjacent leg, [ OA ], in triangle [ OAB ] – or, alternatively, as the ratio of the length of the opposite leg, [ BD ], to the length of the adjacent leg, [ OB ], in triangle [ OBD ], according to (2.297) – once more with the aid of Eq. (2.287), and as emphasized in Fig. 2.10 a; note that Eq. (2.297) may also appear as (2.298) following division of both numerator and denominator by, and with the extra aid of Eqs. (2.288) and (2.290). If θ is expressed in rad, then one has (2.299) in general – as plotted in Fig. 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...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...when t = 1 / b. Returning to the theme of trigonometric applications, recall that the tangent function is a ratio of the sine and cosine functions. The tangent function is periodic (see Figure 7.9A), but unlike the sine and the cosine it is not continuous. Instead, the slope regularly shoots off toward ±∞, with vertical asymptotes at x = π/2, 3π/2, 5π/2, and so on. These asymptotes occur because of the relationship: tan (x) = sin (x) cos (x). (EQ7.14) Box 7.4 Deriving the quotient rule A ratio can be rewritten as a product in which the second term is raised to a negative power, so the equation y = u / v is the same as y = uv −1. We can use the product rule to differentiate this as follows: d y d x = d d x (u v) = d d x (u v − 1) = d u d x × (v − 1) + u × d d x (v − 1). If we set v −1 in the second term on the far right-hand side equal to a new function w, it should become clear that we can differentiate the expression in brackets using the chain rule, because w is a function of v, which itself is a function of x. The differentiation goes as follows: d d x (v − 1) = d w d x = d w d v d v d x = d d x (v − 1) × d v d x = − v − 2 × d v d x. Substituting this into the first expression, we get the following expression for the derivative of y with respect. to x : d y d x = d u d x × (v − 1) − u × v − 2 × d v d x = d u d x × v − u × d v d x v 2. Figure 7.9 Plots of the functions (A) tan(x) and (B) 1/cos 2 (x) for values of x between 0 and 4π. When the graph of the cosine function periodically cuts the x axis, the denominator becomes zero, so the ratio sin(x)/cos(x) becomes infinite. The slope of tan(x) is positive throughout the range 0 ⩽ x ⩽ 2π, but tends to infinity at x = π/2, 3π/2, 5π/2, and so on. These features produce yet another interesting periodic function (see Figure 7.9B) – but how can we quantify its shape? To make progress we need to deploy the quotient rule...
- 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...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART III DERIVATIVES Chapter 7 Derivatives I. DERIVATIVES A. Meaning of Derivative The derivative of a function is its slope. A linear function has a constant derivative since its slope is the same at every point. The derivative of a function at a point is the slope of its tangent line at that point. Non-linear functions have changing derivatives since their slopes (slope of their tangent line at each point) change from point to point. 1. Local linearity or linearization—when asked to find the linearization of a function at a given x -value or when asked to find an approximation to the value of a function at a given x -value using the tangent line, this means finding the equation of the tangent line at a “nice” x -value in the vicinity of the given x -value, substituting the given x -value into it and solving for y. i. For example, approximate using the equation of a tangent line to. We’ll find the equation of the tangent line to at x = 4 (this is the ‘nice’ x -value mentioned earlier). What makes it nice is that it is close to 4.02 and that. Since, so,. Also, f (4) = 2. Substituting these values into the equation of the tangent line, so the equation of the tangent line is. Substituting x = 4.02, y = 2.005. A more accurate answer (using the calculator) is. The linear approximation, 2.005, is very close to this answer. This works so well because the graph and its tangent line are very close at the point of tangency, thus making their y -values very close as well. If you use the tangent line to a function at x = 4 to approximate the function’s value at x = 9, you will get a very poor estimate because at x = 9, the tangent line’s y -values are no longer close to the function’s y -values. ii. The slope of the secant on (a, b), is often used to approximate the value of the slope at a point inside (a, b). For instance, given the table of values of f (x) below, and given that f (x) is continuous and differentiable, approximate f ′(3)...
eBook - ePubSTEM Education by Design
Opening Horizons of Possibility
- Brent Davis, Krista Francis, Sharon Friesen(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...To appreciate this point, it’s useful to pause to think about things in life that repeat themselves in regular cycles. In everyday life, trigonometric functions are probably the most important examples of periodic functions, and perhaps the most familiar of those is the smoothly undulating sine curve, variations of which appear in many of the margins in this chapter. As illustrated in these margin figures, the sine curve can be used to model a great many phenomena, including changes in daylight hours throughout the year and across different latitudes, moon phases and position on the horizon, ocean waves, planetary motion, sound waves, and electromagnetic radiation. Like other mathematics concepts studied in school, this one is important not because it can be used to answer a textbook question about how high a 5 meter ladder will reach if it forms an 80° angle with the ground (although that’s entirely useful!), but because it affords insight into how, for example, making a sound is related to rocking in a boat, playing on a swing, or decreased daylight in winter. The phrasing here is important. Notice that we didn’t say that the sorts of phenomena highlighted in these margins are “examples of the sine function.” Because they aren’t. They are phenomena that can be modeled with the sine function – and there’s a big difference. The sine function isn’t lurking in planetary orbits or sound waves. It is a concept that enables humans to recognize, cluster, and study a particular sort of regularity in the universe. It is a modeling tool. Above and below are several familiar situations that are often modeled using the concept of multiplication. The vital point here is that mathematics is about humanity’s engagement with the world. Concepts are not mined from a mysterious, ideal realm, but are distilled from encounters with many different forms and events. Consider the more familiar concept of multiplication, for example...
