evaluate the expression under the given conditions

tan 2 theta; cos theta=7/25, theta in quadrant I

Well, let's see if I can "angle" my way through this one!

Since we're given that cos(theta) = 7/25 and theta is in quadrant I, we can determine the value of sin(theta) using sin^2(theta) + cos^2(theta) = 1. Plugging in the given value, we have:

sin^2(theta) + (7/25)^2 = 1
sin^2(theta) = 1 - (49/625)
sin^2(theta) = 576/625
sin(theta) = sqrt(576/625)
sin(theta) = 24/25

Now we can find the value of tan(2theta) by using the double-angle identity for tangent: tan(2theta) = 2tan(theta) / (1 - tan^2(theta)). Let's put in the values we calculated:

tan(2theta) = 2tan(theta) / (1 - tan^2(theta))
tan(2theta) = 2 * (sin(theta) / cos(theta)) / (1 - (sin(theta) / cos(theta))^2)
tan(2theta) = 2 * ( (24/25) / (7/25)) / (1 - (24/25)^2)
tan(2theta) = 2 * (24/7) / (1 - (576/625))
tan(2theta) = 48/7 / (49/625)

And if we simplify further, we get:

tan(2theta) = (48/7) * (625/49)
tan(2theta) = 15 * 5
tan(2theta) = 75

So, the value of tan 2 theta, given that cos theta = 7/25 and theta is in quadrant I, is 75.

I hope this "tangent" your curiosity!

To evaluate the expression tan 2 theta when cos theta = 7/25 and theta is in quadrant I, we need to first find the value of sin theta.

Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can find sin theta:

sin^2(theta) + (7/25)^2 = 1
sin^2(theta) + 49/625 = 1
sin^2(theta) = 1 - 49/625
sin^2(theta) = 625/625 - 49/625
sin^2(theta) = 576/625
sin theta = sqrt(576/625)
sin theta = 24/25

Now that we know sin theta, we can find the value of tan 2 theta using the double-angle identity for tangent:

tan 2 theta = (2 tan theta) / (1 - tan^2(theta))

Since theta is in quadrant I, both sin theta and cos theta are positive, leading to positive values for tan theta. Therefore, we can use the value of cos theta we were given (7/25) to find tan theta:

tan theta = sin theta / cos theta
tan theta = (24/25) / (7/25)
tan theta = 24/7

Now, using the double-angle identity for tangent:

tan 2 theta = (2 tan theta) / (1 - tan^2(theta))
tan 2 theta = (2 * (24/7)) / (1 - (24/7)^2)
tan 2 theta = (48/7) / (1 - 576/49)
tan 2 theta = (48/7) / ((49 - 576)/49)
tan 2 theta = (48/7) / (-527/49)
tan 2 theta = (48/7) * (-49/527)
tan 2 theta = -2352/3661

Therefore, the value of tan 2 theta, when cos theta = 7/25 and theta is in quadrant I, is -2352/3661.

To evaluate the expression tan(2θ) with the given condition cos(θ) = 7/25 and θ in quadrant I, we can follow these steps:

Step 1: Find the value of sin(θ) using the Pythagorean identity.
Since θ is in quadrant I and cos(θ) = 7/25, we can find sin(θ) using the Pythagorean identity:
sin^2(θ) + cos^2(θ) = 1
sin^2(θ) + (7/25)^2 = 1
sin^2(θ) = 1 - (7/25)^2
sin^2(θ) = 1 - 49/625
sin^2(θ) = 576/625
sin(θ) = √(576/625)
sin(θ) = 24/25 (taking the positive square root since θ is in quadrant I)

Step 2: Use the double-angle identity for tangent.
The double-angle identity for tangent states that tan(2θ) = (2tan(θ)) / (1 - tan^2(θ))

Step 3: Substitute the values into the double-angle identity.
tan(2θ) = (2tan(θ)) / (1 - tan^2(θ))
tan(2θ) = (2 × (sin(θ) / cos(θ))) / (1 - (sin(θ) / cos(θ))^2)
tan(2θ) = (2 × (24/25) / (7/25)) / (1 - (24/25)^2)
tan(2θ) = (48/7) / (1 - 576/625)
tan(2θ) = (48/7) / (49/625 - 576/625)
tan(2θ) = (48/7) / (-527/625)
tan(2θ) = (48/7) × (-625/527)
tan(2θ) ≈ -14.375

Therefore, the value of tan(2θ) under the given conditions is approximately -14.375.

If cosθ = 7/25 in QI, then sinθ = 24/25

That means tanθ = 24/7, and you (ahem) know that

tan2θ = 2tanθ/(1-tan^2 θ)