Derivative of Trigonometric Functions
The derivative of trigonometric functions involves finding the rate of change of these functions with respect to the input variable. For example, the derivative of sine is cosine, and the derivative of cosine is negative sine. These derivatives are essential in calculus for solving problems involving angles, periodic motion, and oscillations.
4 Key excerpts on "Derivative of Trigonometric Functions"
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...
- 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 - ePub- Nicholas L. Georgakopoulos(Author)
- 2018(Publication Date)
- Palgrave Macmillan(Publisher)
...© The Author(s) 2018 Nicholas L. Georgakopoulos Illustrating Finance Policy with Mathematica Quantitative Perspectives on Behavioral Economics and Finance https://doi.org/10.1007/978-3-319-95372-4_3 Begin Abstract 3. The Mathematical Frontier: Trigonometry, Derivatives, Optima, Differential Equations Nicholas L. Georgakopoulos 1 (1) Indiana University, Indianapolis, IN, USA Nicholas L. Georgakopoulos Email: [email protected] End Abstract Graphical applications bring up trigonometrical functions, such as sine and cosine, frustratingly often. This chapter uses Mathematica’s graphics to review the basics of trigonometry. Being in a review mode, the chapter continues to review derivatives. We close with our first legal application, differential equations applied to legal change. 1 Trigonometry Illustrated To illustrate trigonometry, begin by creating a circle with radius one and with its center at the origin of the coordinate system, i.e., at the point (0, 0). Use Mathematica’s Circle[] command. Its syntax is Circle[{ x-coord, y-coord }, radius ] but radius is set at one if omitted. Graphics[Circle[{0,0}], Axes->True] produces the desirable outcome. 1.1 An Angle, Its Sine, and Cosine Form an angle using the horizontal axis, the x-axis, and a line that intersects the axis at the origin. Measure the angle counterclockwise from the positive values of the x-axis. The sine of this angle is the height, or y -coordinate, of the point where the line that intersects the axis crosses the circle. The cosine of the same angle is the x -coordinate of the same point. Thus, we can place this point by using the Point[ ] command. Point[{Cos[ angle ], Sin[ angle ]}] would produce that result. To emphasize the point, increase its size by preceding it with a PointSize[ ] command. Form the angle by drawing the line that intersects the x-axis. Figure 1 shows my resulting screen. Fig...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...For the example above, one would have been led to (2.284) after setting m = 3 and n = 2 in Eq. (2.282), while Eq. (2.283) would yield (2.285) the possibilities of integer values for (k 1, k 2, k 3) satisfying the condition placed at the bottom of the summation in Eq. (2.284) encompass (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), and (0,1,1) – so Eq. (2.284) becomes (2.286) with the aid of Eq. (2.285). After realizing that 0! = 1, 2!/2! = 1, and 2!/1! = 2 – besides – one eventually retrieves Eq. (2.281), using Eq. (2.286) as departure point. 2.3 Trigonometric Functions Trigonometry is the branch of mathematics that studies relationships involving lengths of sides and amplitudes of angles in triangles. This field emerged in the Hellenistic world during the third century, by the hand of Euclid and Archimedes – who studied the properties of chords and inscribed angles in circles, while proving theorems equivalent to most modern trigonometric formulae; Hipparchus from Nicaea (Asia Minor) produced, however, the first tables of chords in 140 BCE – analogous to the current tables of sine values, which were completed in the second century CE by Greco‐Egyptian astronomer Ptolemy from Alexandria (Egypt). By those times, it was realized that the lengths of the sides of a right triangle and the angles between those sides satisfy fixed relationships; hence, if at least the length of one side and the amplitude of one angle is known, then all other angles and lengths can be algorithmically determined. 2.3.1 Definition and Major Features Consider a unit vector u, i.e. (2.287) with double bars indicating length, centered at the origin of a system of coordinates, which rotates around said origin – as illustrated in Fig. 2.10 a...
