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.