Answer:
Given that sin(θ) = √65/9 and θ is in the second quadrant, we can find the exact value of cos(θ) using the Pythagorean identity: sin²(θ) + cos²(θ) = 1.
First, let's find cos²(θ):
cos²(θ) = 1 - sin²(θ)
cos²(θ) = 1 - (√65/9)²
cos²(θ) = 1 - 65/81
cos²(θ) = (81 - 65)/81
cos²(θ) = 16/81
Taking the square root of both sides to find cos(θ):
cos(θ) = ±√(16/81)
cos(θ) = ±(4/9)
Since θ is in the second quadrant where cosine is negative, we choose the negative value:
cos(θ) = -(4/9)
Now, let's answer the other questions:
The domain of f(x) = cos(x) is all real numbers. In other words, you can input any real number as x in the function.
The range of f(x) = cos(x) is [-1, 1]. The cosine function oscillates between -1 and 1 and takes on all values within this range as x varies.