Final answer:
To find the exact value of cos(2θ) given cos(0) = -4/5 and θ is in quadrant 2, we can use the double angle identity cos(2θ) = cos²(θ) - sin²(θ). By determining the values of cos(θ) and sin(θ) using the given information, we can substitute them into the double angle identity to find the exact value of cos(2θ), which is 7/25.
Step-by-step explanation:
To find the exact value of cos(2θ), we can use the double angle identity cos(2θ) = cos²(θ) - sin²(θ).
Given that cos(0) = -4/5 and θ is in quadrant 2, we can determine the values of cos(θ) and sin(θ). Since cos(0) = -4/5, we know that cos(θ) is also negative in quadrant 2. Let's assume cos(θ) = -4/5. To find sin(θ), we can use the Pythagorean identity sin²(θ) = 1 - cos²(θ). Plugging in the value of cos(θ) = -4/5, we get sin²(θ) = 1 - (-4/5)² = 1 - 16/25 = 9/25. Taking the square root of both sides, we find sin(θ) = 3/5.
Now we can substitute the values of cos(θ) = -4/5 and sin(θ) = 3/5 into the double angle identity cos(2θ) = cos²(θ) - sin²(θ). Plugging in these values, we have cos(2θ) = (-4/5)² - (3/5)² = 16/25 - 9/25 = 7/25.
Therefore, the exact value of cos(2θ) is 7/25.