Final answer:
The solution for cotθ=0.7615 does not correspond to any of the provided options. Through calculation, the angles are found to be approximately 0.9236 radians (first quadrant) and 4.0649 radians (third quadrant), which do not match any of the options given.
Step-by-step explanation:
To solve cotθ=0.7615 on the interval 0≤θ≤2π, we need to find the angles where the cotangent value is 0.7615. Cotangent is the reciprocal of tangent, meaning cotθ = 1/tanθ. We would typically start by finding the arctangent of the reciprocal of 0.7615 to get the corresponding angle.
Let's calculate the arctangent: θ = arctan(1/0.7615). Using a calculator, we find that θ ≈ arctan(1.3130), which gives us θ ≈ 52.9185° in the first quadrant. However, cotangent is positive in both the first and third quadrants, so we also need to find the angle in the third quadrant which will be θ ≈ 180° + 52.9185° = 232.9185°. Converting these degree values to radians, we have:
θ ≈ 0.9236 radians (first quadrant) and θ ≈ 4.0649 radians (third quadrant).
Comparing these angles to the given options:
- a. θ=π/4 is approximately 0.7854 radians, which does not match.
- b. θ=3π/4 is approximately 2.3562 radians, which does not match.
- c. θ=5π/4 is approximately 3.9269 radians, which does not match.
- d. θ=7π/4 is approximately 5.4978 radians, which does not match.
Therefore, none of the given options (a, b, c, or d) correspond to the correct angles found for cotθ=0.7615.