arctan(tan(2pi/3)
thanks.
arctan(tan(2pi/3) = -pi/3
since arctan and tan are inverse operations, the solution would be 2pi/3
the number of solutions to arctan(x) is infinite, look at its graph.
generally, unless a general solution is asked for, or a domain is specified,
arc(trigratio) answers are given in the domain 0 <-----> 2pi
Using the CAST rule, two solutions are then possible, 2pi/3 and 5pi/3
when punched in a calculator, your machine will blindly give you -pi/3, because it cannot do the analysis, and is programmed to give the closest answer to zero.
I have to disagree with Reiny here. The arctan function is conventionally defined (and you have to choose some definition) such that arctan(tan(x) = x when x is between -pi/2 to pi/2.
Given this definition of arctan, the answer is -pi/3. Note that you are applying some given function to the tan and that function does not know what went in the tan.
A similar case is that of the squareroot function. One conventionally defines the squareroot to be the positve root, not the negative root. So, it's wrong to say that
sqrt[(-1)^2] = -1
The inverse trignometric functions, square roots etc. can all be expressed as logarithms. Once a definition of the logarithm is chosen, e.g. by putting the branch cut on the negative real axis, all the other functions are defined.
It is very easy to make mistakes by using inconsistent definitions of the inverse trignometric function. There is a well known example in theoretical physics were some anomalous behavior was predicted for some capture cross section. Years later it transpired that it was all due to a simple artifact of wrongly defined inverse functions.
To find the value of arctan(tan(2pi/3), we need to understand the relationship between arctan and tan functions.
The tangent function (tan) relates the ratio of the length of the opposite side to the adjacent side of a right triangle, while the arctan function (also known as the inverse tangent) gives us the angle that corresponds to a given tangent ratio.
In this case, we want to find the angle whose tangent is equal to tan(2pi/3).
To do so, we need to understand the characteristics of the tangent function. The tangent function is periodic, with a period of pi. Therefore, any angle x and x + n*pi (where n is an integer) will have the same tangent value.
In the given expression, tan(2pi/3) equals the tangent of an angle that lies in the second quadrant.
To find the value of arctan(tan(2pi/3)), we can follow these steps:
Step 1: Find the value of tan(2pi/3)
Using a calculator or reference table, we can calculate the value of tan(2pi/3) to be approximately -1.732.
Step 2: Determine the angle whose tangent is -1.732
Since -1.732 is negative, it lies in the second and fourth quadrants.
Step 3: Find the reference angle
The reference angle is the acute angle formed between the horizontal axis and the terminal side of the angle formed with the given tangent. To find the reference angle, we can use the inverse tangent (arctan) function, which gives us the angle in the range of -pi/2 to pi/2.
arctan(1.732) is approximately 1.047.
Step 4: Find the solution in the given quadrant
Since we are interested in the second quadrant, we need to add pi radians (180 degrees) to the reference angle found in step 3.
1.047 + pi = approximately 4.189 radians.
Therefore, arctan(tan(2pi/3)) is approximately 4.189 radians, or approximately 120.69 degrees.
It's important to note that the arctan function has infinite solutions, so there are other angles that have the same tangent value. However, within the domain of 0 to 2pi (or 0 to 360 degrees), the solution we found is the one within that range.
For this specific problem, it's crucial to follow the convention of arctan being defined in the range of -pi/2 to pi/2.