if 25 cos ceta = 7 where ceta belongs to ] 3 pi/2 , 2 pi [ , then tan ceta =

Answer 1

To find the value of tan(theta), we need to first solve for theta using the equation 25 cos(theta) = 7.

First, divide both sides of the equation by 25:
cos(theta) = 7/25

Next, find the angle whose cosine is 7/25 within the given range.

Since cos(theta) is positive, we know that theta will be in the first or fourth quadrant.

The inverse cosine of 7/25 is approximately 1.067.
Therefore, theta ≈ 1.067.

Now, let's find the tangent of theta.

We know that tan(theta) = sin(theta) / cos(theta).

Since we already have the value of cos(theta) and cos(theta) is positive, we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1

By substituting cos(theta) = 7/25, we can solve for sin(theta):
sin^2(theta) + (7/25)^2 = 1
sin^2(theta) + 49/625 = 1
sin^2(theta) = 1 - 49/625
sin^2(theta) = 576/625

Taking the square root of both sides, we get:
sin(theta) = ± 24/25

Since theta is in the interval ]3pi/2, 2pi[, sin(theta) is negative.

So, sin(theta) = -24/25.

Now, we can finally calculate the tangent of theta:
tan(theta) = sin(theta) / cos(theta)
tan(theta) = (-24/25) / (7/25)
tan(theta) = -24/7

Therefore, tan(theta) = -24/7.