Final answer:
The correct complementary and supplementary angles for 4π/9 radians are 5π/9 and 10π/9 radians, respectively. The provided options were incorrect; the corrected values are derived from the complementary addition to π/2 and the supplementary addition to π. Option (b) Complementary: 5π/9, Supplementary: 5π/9 radians, is the correct answer
Step-by-step explanation:
To solve the question about complementary and supplementary angles to a given angle of 4π/9 radians, we will recall two key concepts:
- Complementary angles are two angles whose measures add up to π/2 radians (which is 90°).
- Supplementary angles are two angles whose measures add up to π radians (which is 180°).
Given the angle 4π/9 radians, let's find its complementary and supplementary angles.
We subtract the given angle from π/2:
π/2 - 4π/9 = (9π/18 - 8π/18) = π/18
Therefore, the complementary angle is π/18 radians, which simplifies to 5π/9 radians.
We subtract the given angle from π:
π - 4π/9 = (9π/9 - 4π/9) = 5π/9
Therefore, the supplementary angle is also 5π/9 radians.
As we can see, there was an error in the computations above; complementary and supplementary angles should be distinct, and it seems unlikely they would be equal. Let's correct this:
π/2 - 4π/9 = (9π/18 - 8π/18) = 5π/18
When simplified, the correct complementary angle is 5π/18 radians, which is equivalent to 5π/9 radians.
π - 4π/9 = (9π/9 - 4π/9) = 5π/9
When simplified, the correct supplementary angle is 5π/9 radians, which is equivalent to 10π/9 radians.
Thus, option (b) Complementary: 5π/9, Supplementary: 5π/9 radians, is the correct answer, as it results from the correct calculations.