Answer: csc(x) = (√b) / √(b(b - a))
Explanation:
To find the cosecant (csc) of an angle x when you know the cosine (cos) of that angle, you can use the following trigonometric identity:
csc(x) = 1 / sin(x)
First, you need to find the sine (sin) of x. Since you know that cos(x) = √(a/b), you can use the Pythagorean identity to find sin(x):
sin(x) = √(1 - cos^2(x))
sin(x) = √(1 - (√(a/b))^2)
sin(x) = √(1 - a/b)
sin(x) = √((b - a) / b)
Now that you have sin(x), you can find csc(x):
csc(x) = 1 / sin(x)
csc(x) = 1 / (√((b - a) / b))
To rationalize the denominator, you can multiply both the numerator and denominator by √b:
csc(x) = (√b) / (√b * √((b - a) / b))
csc(x) = (√b) / √(b(b - a))
So, the cosecant of x is:
csc(x) = (√b) / √(b(b - a))
This is the expression for the cosecant of x in terms of a and b when the cosine of x is √a/b.