Tan( theta) = 3/4 where pi<theta<3pi/2. Find the exact value of cos(2theta) using those values.

Isn't theta a 3.4.5 triangle? If so, you can use your double angle formula.

To find the exact value of cos(2theta), we can use the double angle formula for cosine:

cos(2theta) = cos^2(theta) - sin^2(theta)

First, let's find the value of sin(theta). We know that tan(theta) = 3/4. Since tan(theta) = sin(theta) / cos(theta), we can set up the equation:

3/4 = sin(theta) / cos(theta)

Cross-multiplying, we have:

4sin(theta) = 3cos(theta)

Squaring both sides to eliminate the square root, we get:

16sin^2(theta) = 9cos^2(theta)

Using the identity sin^2(theta) + cos^2(theta) = 1, we can rewrite the equation as:

16(1 - cos^2(theta)) = 9cos^2(theta)

Expanding and rearranging terms, we have:

16 - 16cos^2(theta) = 9cos^2(theta)

Combining like terms, we get:

25cos^2(theta) = 16

Dividing both sides by 25, we find:

cos^2(theta) = 16/25

Taking the square root of both sides, we get:

cos(theta) = ±4/5

Since theta is in the third quadrant (pi < theta < 3pi/2), the cosine is negative. Therefore, cos(theta) = -4/5.

Now that we have the value of cos(theta), we can find sin(theta) by rearranging the equation sin(theta) = (4/5) * cos(theta):

sin(theta) = (4/5) * (-4/5) = -16/25

Now we can compute the value of cos(2theta) using the double angle formula:

cos(2theta) = cos^2(theta) - sin^2(theta)

Substituting the values we found earlier:

cos(2theta) = (-4/5)^2 - (-16/25)^2

Simplifying:

cos(2theta) = 16/25 - 256/625

Finding a common denominator:

cos(2theta) = (400 - 256) / 625

cos(2theta) = 144/625

Therefore, the exact value of cos(2theta) is 144/625.