Write an algebraic expression in x for sec(arcsin(x/sqrt(x^(2) +4)))
sin A = x/sqrt(x^2+4)
draw that right triangle
side adjacent to A is
sqrt (x^2+4-x^2) = sqrt 4 = 2
so
cos A = 2/sqrt(x^2+4)
sec A = 1/cos A = sqrt(x^2+4)/2
To express sec(arcsin(x/sqrt(x^(2) +4))) algebraically in terms of x, we need to simplify the given expression step by step.
Let's start by looking at the innermost expression inside the arcsin function, which is x/sqrt(x^(2) +4). We can label this expression as y for simplicity.
So, y = x/sqrt(x^(2) + 4).
Now, to find the value of y that will allow us to use the arcsin function, we need to consider the domain of arcsin, which is -1 ≤ y ≤ 1.
Looking at the expression y = x/sqrt(x^(2) + 4), we can see that for x ≠ 0, the denominator will always be greater than zero. Therefore, the expression is never undefined, and the domain of y covers all real numbers.
Next, we need to determine the value of arcsin(y). Since the domain of y covers all real numbers, the arcsin function will output an angle between -π/2 and π/2, inclusive.
Now, we need to express sec(arcsin(y)) algebraically using trigonometric identities.
To do that, we recall the identity: sec(theta) = 1/cos(theta).
Since our angle is represented as arcsin(y), we need to find the value of cosine relating to this angle.
To find the cosine of the angle, we can use the Pythagorean identity, sin^(2)(theta) + cos^(2)(theta) = 1.
Since arcsin(y) is our angle, we can rewrite it as sin(theta) = y.
Plugging this into the Pythagorean identity, we have:
y^(2) + cos^(2)(theta) = 1.
Solving for cos(theta), we get:
cos(theta) = sqrt(1 - y^(2)).
So, sec(arcsin(y)) becomes:
sec(arcsin(y)) = 1/cos(arcsin(y)) = 1/sqrt(1 - y^(2)).
Finally, substituting y with x/sqrt(x^(2) + 4) in sec(arcsin(y)), we obtain the algebraic expression in x for sec(arcsin(x/sqrt(x^(2) +4))):
sec(arcsin(x/sqrt(x^(2) + 4))) = 1/sqrt(1 - (x/sqrt(x^(2) + 4))^(2)).
Simplifying this expression further is possible, but this is the final form of the algebraic expression for sec(arcsin(x/sqrt(x^(2) +4))).