Question

Given cos(θ) = 7/9 and sin(θ) < 0, calculate the exact value of cot(θ).

a) -√2/3
b) -3/√2
c) -√2/7
d) -7/√2

Answer 1

Given cos(θ) = 7/9 and sin(θ) < 0, by using the Pythagorean identity, we find sin(θ) = -4√(2)/9. We then calculate cot(θ) = cos(θ)/sin(θ), resulting in cot(θ) = -7√(2)/8, which is not listed in the given answer choices.

Step-by-step explanation:

To find the exact value of cot(θ) given cos(θ) = 7/9 and sin(θ) < 0, we first need to determine the value of sin(θ). Since we are given cos(θ), we can use the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1.

Now, we calculate sin(θ):

• cos²(θ) = (7/9)² = 49/81
• sin²(θ) = 1 - cos²(θ) = 1 - 49/81 = 32/81
• sin(θ) = ±√(32/81)
• Since sin(θ) < 0, we take the negative square root: sin(θ) = -√(32)/9 = -4√(2)/9

Then, the cotangent of θ, which is cot(θ) = cos(θ)/sin(θ), is calculated as follows:

• cot(θ) = (7/9) / (-4√(2)/9)
• cot(θ) = -7/4√(2)
• Multiply by √(2)/√(2) to rationalize the denominator
• cot(θ) = -7√(2)/8

Therefore, the exact value of cot(θ) is -7√(2)/8, which is not one of the answer choices provided. The calculation here might suggest a possible mistake in the given answer options or the initial conditions of the problem.