Find all solutions of the equation (sec(x))^2−2=0
The answer is A+Bk where k is any integer and 0<A<pi/2
Find:
A= pi/4?
B=??
(sec x)^2 = 2
(cos x)^2 = 1/2
cos x = ± 1/√2
we know cos π/4 = 1/√2
so x could be 45°, 135°, 225° , 315° ...
notice we add 90° each time
so the general solution is
π/4 + π/2 k, where k is an integer
matching this with A + Bk
A = π/4
B = π/2
To solve the equation (sec(x))^2 - 2 = 0, we can start by isolating the squared secant term:
(sec(x))^2 = 2
Now, taking the square root of both sides, we have:
sec(x) = ±√2
Recall that the secant function is the reciprocal of the cosine function, so we can rewrite the equation as:
1/cos(x) = ±√2
To find the solutions, we need to consider the possible values of cos(x) that satisfy the equation.
When cos(x) = 1/√2, the reciprocal would be 1 / (1/√2) = √2, which satisfies the equation:
cos(x) = 1/√2 => x = π/4 + 2kπ where k is any integer
Similarly, when cos(x) = -1/√2, the reciprocal would be 1 / (-1/√2) = -√2, which also satisfies the equation:
cos(x) = -1/√2 => x = 3π/4 + 2kπ where k is any integer
In both cases, we have found values of x that satisfy the given equation.
Therefore, the solutions to (sec(x))^2 - 2 = 0 are:
x = π/4 + 2kπ where k is any integer
x = 3π/4 + 2kπ where k is any integer
Now, let's find the specific values for A and B mentioned in the answer.
From the solutions, we can see that A represents the constant term π/4 (the non-transformed part of the angle) when k = 0.
So, A = π/4.
On the other hand, B represents the coefficient of the integer k when k ≠ 0.
Since k can be any integer, it is represented by B. If multiple values of k are needed, B would be 1. However, if only one solution is needed, B would be 0.
Therefore, the values are:
A = π/4
B = 0 or 1 (depending on the context of the problem)