The given figure shows a circle centered at the origin. If the arc 𝐵𝐶𝐷 has a length of 27𝜋 minus four over three, find the measure of angle 𝐴𝑂𝐵 in radians.
This is the arc 𝐵𝐶𝐷. And this is the angle we’re interested in, 𝐴𝑂𝐵. We know that the arc length is 27𝜋 minus four over three. But the information we’ll need to solve this problem is the angle measure that creates this arc. We know that an arc length 𝑠 is equal to the angle that creates that arc times the radius. And so if we wanted to calculate this angle 𝜃, we would take the arc length and divide it by the radius. And because we know that the circle is centered at the origin and that 𝐶 is located at six, zero, we can say that the radius is six units. And that means 𝜃 will be equal to the arc length 27𝜋 minus four divided by three, divided by the radius of six. We can rewrite that to say 27𝜋 minus four over three times one-sixth. Dividing by six is the same thing as multiplying by one over six. And so we’ll multiply those denominators together and the numerators together. 27𝜋 minus four over 18 is the angle measure that creates the arc 𝐵𝐶𝐷.
We now need to think about something else we know about this circle. If we add angle 𝐴𝑂𝐵 and angle 𝜃, which is angle 𝐵𝑂𝐷, then we would get this space. This is three-quarter turns, 90 degrees times three. We need to think about what 270 degrees would be in radians. I know that 180 degrees equals 𝜋 radians. And that 90 degrees is 𝜋 over two radians. This means that 270 degrees would be equal to 𝜋 plus 𝜋 over two radians. And that is three 𝜋 over two. If we take this 270 degrees, this three 𝜋 over two radians, and subtract angle 𝜃, angle 𝐵𝑂𝐷, it will give us the measure of angle 𝐴𝑂𝐵. That means we need to take three 𝜋 over two and subtract 27𝜋 minus four over 18.
First, we need to calculate a common denominator between the two fractions. If we multiply three 𝜋 over two times nine over nine, these two fractions will have a common denominator of 18. Nine times three 𝜋 equals 27𝜋. And nine times two equals 18 minus 27𝜋 minus four over 18. Now that they have a common denominator, we can add and subtract in the numerator. Look carefully at the numerator. It is now 27𝜋 minus 27𝜋 plus four. We were subtracting negative four. And that became positive four. And then we can say that 27𝜋 minus 27𝜋 cancels out. It’s equal to zero. So we have angle 𝐴𝑂𝐵 will be equal to four over 18. And we wanna give it in its simplified form, which is two-ninths. Angle 𝐴𝑂𝐵 measures two-ninths radians. Angle 𝜃, angle 𝐵𝑂𝐷, measures 27𝜋 minus four over 18 radians. Together, these two angles measure three 𝜋 over two radians.
We’re only interested in angle 𝐴𝑂𝐵. And that is two-ninths radians.