What is Arcsec(-2) equal to and how can you find the correct answer? I got 2pi/3 but it was marked wrong on my test and I'm not sure why it's wrong. Can you please explain this to me? Thanks!
sec = 1/cos
so what angle has cosine = -1/2
that angle is 180 + or - 60 degrees
which is pi +/- pi/3
2 pi/3 or 4 pi/3
arcsec(-2) is the same angle as arccos(-1/2)
The angle 2pi/3 is correct, so is 4pi/3.
Depending on the domain required, you may have to put in both values (for 0-2pi).
If that is not the problem, then you have to read carefully the instructions how to answer the question, how to type in the %pi; symbol, or the number of significant figures, etc.
since cos pi/3 = 1/2,
sec pi/3 = 2
But, arcsec has principal values from 0 to pi, so arcsec(-2) = pi - pi/3 = 2pi/3.
Note: Some authors define the range of arcsecant to be ( 0 ≤ x < π/2 or π ≤ x < 3π/2 ), because the tangent function is nonnegative on this domain. This makes some computations more consistent.
So, you'd better check to see how your course defines the principal values. Apparently you were expected to provide 4pi/3 as the arcsec(-2)
Perhaps you were supposed to give both the quadrant 2 and quadrant 2 answers
Oh, thank you so much Steve. I just found a very old sheet from class with the restrictions and you are right-- my teacher gave the unconventional one. I studied from online sources, which all gave the other range, so that's why I got it wrong. Thanks!
Arcsec(-2) represents the arcsecant function of -2. To find the correct answer, we need to understand what arcsecant means.
The arcsecant (or arccosecant) function, denoted as arcsec(x) or sec^(-1)(x), is the inverse of the secant function (sec(x)). In other words, for any given value of y, arcsec(y) gives us the angle whose secant is y.
To find the value of arcsec(-2), we need to find the angle whose secant is -2. Since the secant function is the reciprocal of the cosine function, we can rephrase the question as finding the angle θ whose cosine is -1/2 (since sec(x) = 1/cos(x)).
To do this, we need to recall the unit circle and the values of cosine for different angles. At θ = π/3, the cosine is indeed -1/2. However, it's important to note that the secant function is defined as the reciprocal of cosine only in the interval [-π/2, 0) and (0, π/2].
Beyond these intervals, the secant function repeats its values. In particular, since the cosine is symmetric around the y-axis, the secant function is symmetric around the origin (0,0), resulting in a periodic pattern. When the secant function repeats its values, the arcsecant function also repeats its values but with a positive sign instead of negative.
Thus, we have two solutions for arcsec(-2): one in the interval [-π/2, 0) and another in the interval (0, π/2].
In the interval [-π/2, 0), the corresponding angle θ is π - π/3 = 2π/3. So your initial solution of 2π/3 is correct in this interval.
In the interval (0, π/2], we need to find an angle such that its cosine is -1/2. Here, we should refer to θ = π/3, which has a positive cosine value of 1/2. The symmetric angle with respect to the origin (-π/3) will have a negative cosine value of -1/2. Thus, the correct answer within this interval is -π/3.
Therefore, the correct and complete solution for arcsec(-2) is both 2π/3 and -π/3.