105k views
5 votes
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

User Craken
by
8.2k points

1 Answer

2 votes

Final answer:

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.

User RayofCommand
by
8.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.