(Math | Calculus | Derivatives | Table Of)
|sin x = cos x Proof||csc x = -csc x cot x Proof|
|cos x = - sin x Proof||sec x = sec x tan x Proof|
|tan x = sec2 x Proof||cot x = - csc2 x Proof|
Similarly, you may ask, what is the tangent of 0?
So let's take a closer look at the sine and cosines graphs, keeping in mind that tan(θ) = sin(θ)/cos(θ). The tangent will be zero wherever its numerator (the sine) is zero. This happens at 0, π, 2π, 3π, etc, and at –π, –2π, –3π, etc.
How do you find the equation of a tangent line?
1) Find the first derivative of f(x). 2) Plug x value of the indicated point into f '(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line.
The derivative of tan x is sec2x. However, there may be more to finding derivatives of tangent. In the general case, tan x is the tangent of a function of x, such as tan g(x).
DIFFERENTIATION USING THE CHAIN RULE. The following problems require the use of the chain rule. The chain rule is a rule for differentiating compositions of functions. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Most problems are average.
Cosecant, Secant, and Cotangent. In addition to sine, cosine, and tangent, there are three other trigonometric functions you need to know for the Math IIC: cosecant, secant, and cotangent. These functions are simply the reciprocals of sine, cosine, and tangent. Cosecant. Cosecant is the reciprocal of sine.
Implicit differentiation is nothing more than a special case of the well-known chain rule for derivatives. The majority of differentiation problems in first-year calculus involve functions y written EXPLICITLY as functions of x . For example, if. , then the derivative of y is.
To differentiate f(x)= sin(2x), simply use the chain rule. Namely, dy/dx= 2*cos(2x). The derivative of the sin(x) with respect to x is the cos(x), and the derivative of 2x with respect to x is simply 2.
The trig function cotangent, written cot θ. cot θ equals or . For acute angles, cot θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = cot x is a periodic function with period π.
f(x) = (sin x)2 can be written as f(u) = u2 where u = sin x. The Chain Rule states that to differentiate a composite function we differentiate the outer function and multiply by the derivative of the inner function. We can use the rules cos x = sin ( /2– x) and sin x = cos( /2 – x) to find the derivative of cos x.
A Common Service Center (CSC) is an information and communication technology (ICT) access point created under the National e-Governance Project of the Indian government. The project plan includes the creation of a network of over 100,000 CSCs throughout the country.
The derivative of cos(2x) is -2sin(2x). The process of finding this derivative uses the chain rule. We can use integrals to check our work when finding derivatives. If D(x) is the derivative of f(x), then the integral of D(x) is f(x) + C, where C is a constant.
So far in this course, the only trigonometric functions which we have studied are sine and cosine. The tangent of x is defined to be its sine divided by its cosine: tan x = sin x cos x . The cotangent of x is defined to be the cosine of x divided by the sine of x: cot x = cos x sin x .
Thus we sometimes say that the antiderivative of a function is a function plus an arbitrary constant. Thus the antiderivative of cos x is (sin x) + c. The more common name for the antiderivative is the indefinite integral.
Secant, cosecant and cotangent, almost always written as sec, cosec and cot are trigonometric functions like sin, cos and tan. Note, sec x is not the same as cos-1x (sometimes written as arccos x). Remember, you cannot divide by zero and so these definitions are only valid when the denominators are not zero.
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let where both and are differentiable and. The quotient rule states that the derivative of is.
The arctangent of x is defined as the inverse tangent function of x when x is real (x∈ℝ). When the tangent of y is equal to x: tan y = x. Then the arctangent of x is equal to the inverse tangent function of x, which is equal to y: arctan x= tan-1 x = y.
Here are two helpful hints: Each of those definitions has a cofunction on one and only one side of the equation, so you won't be tempted to think that sec A equals 1/sin A. And secant and cosecant go together just like sine and cosine, so you won't be tempted to think that cot A equals 1/sin A.
When the cosine of y is equal to x: cos y = x. Then the arccosine of x is equal to the inverse cosine function of x, which is equal to y: arccos x = cos-1 x = y. (Here cos-1 x means the inverse cosine and does not mean cosine to the power of -1).
Trigonometry/Cosh, Sinh and Tanh. The functions cosh x, sinh x and tanh x have much the same relationship to the rectangular hyperbola y2 = x2 - 1 as the circular functions do to the circle y2 = 1 - x2. They are therefore sometimes called the hyperbolic functions (h for hyperbolic).
The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot). It is often simpler to memorize the the trig functions in terms of only sine and cosine: sin(q) = opp/hyp. csc(q) = 1/sin(q)
Secant (sec) - Trigonometry function. (See also Secant of a circle). In a right triangle, the secant of an angle is the length of the hypotenuse divided by the length of the adjacent side.
cotangent(q) = adj/opp. Furthermore, the functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).