Find a numerical value of one trigonometric function of x if secxcotx=4
A.cscx=1/4
b.secx=4
c.secx=1/4
d.cscx=4
secxcotx=4
(1/cosx)(cosx/sinx) = 4
1/sinx = 4
cscx = 1/4
To find the numerical value of one trigonometric function of x, we need to solve the given equation and then substitute the solution into the trigonometric function.
We are given:
sec(x) * cot(x) = 4
Using trigonometric identities, we can rewrite cot(x) as 1/tan(x) and sec(x) as 1/cos(x):
(1/cos(x)) * (1/tan(x)) = 4
Multiplying both sides by cos(x) and tan(x):
1 * (1/cos(x)) = 4 * cos(x) * tan(x)
Simplifying:
1/cos(x) = 4 * cos(x) * sin(x) / cos(x)
1/cos(x) = 4 sin(x)
Now, we can use the reciprocal identity for sine to rewrite sin(x) as 1/csc(x):
1/cos(x) = 4 / csc(x)
Substituting csc(x) with its reciprocal form:
1/cos(x) = 4 * sin(x)
Next, we can use the Pythagorean identity for sine and cosine: sin^2(x) + cos^2(x) = 1
Substituting 1 - sin^2(x) for cos^2(x):
1/cos(x) = 4 * √(1 - (1/cos^2(x)))
To simplify further, we can square both sides of the equation:
(1/cos(x))^2 = (4 * √(1 - (1/cos^2(x))))^2
1/cos^2(x) = 16(1 - (1/cos^2(x)))
Multiplying both sides by cos^2(x):
1 = 16cos^2(x) - 16
Rearranging the equation:
16cos^2(x) = 17
Dividing by 16:
cos^2(x) = 17/16
Taking the square root of both sides:
cos(x) = ±√(17/16)
Since sec(x) is the reciprocal of cos(x), we can find its value by taking the reciprocal of cos(x):
sec(x) = 1/cos(x)
Therefore, the numerical value of sec(x) is ±√(16/17).
From the answer choices, option B is the correct answer: sec(x) = 4.