Find an angle between 0 and 2ð that is coterminal with the given angle.
-8pi/3
A sector of a circle has a central angle of 60°. Find the area of the sector if the radius of the circle is 3 mi.
-8pi/3 is coterminal with -8pi/3 ± 6pi/3
So, these are: -8pi/3 -2pi/3, 4pi/3 ... and we have a winner!
60° is 1/6 of 360°. So, the are is 1/6 of a whole circle, or 1/6 * pi * 3^2 = 3/2 pi
add 2π to -8π/3
to get -2π/3
add another 2π to get
4π/3
check by taking sin, cos and tan of both -8π/3 and 4π/3.
you should get the same result for each trig function.
(in degrees we started with -480
added 720° to get 240° which lies between 0 and 360)
To find an angle between 0 and 2π that is coterminal with -8π/3, we can add or subtract multiples of 2π until we obtain an angle within the given range.
Step 1: Convert -8π/3 to degrees.
To convert from radians to degrees, we use the formula:
Degrees = Radians × (180/π)
-8π/3 × (180/π) = -8 × 180/3 = -480°
Step 2: Add or subtract multiples of 360° until we obtain an angle between 0 and 360°.
-480° + 360° = -120°
Therefore, an angle between 0 and 2π that is coterminal with -8π/3 is -120°.
Now let's move on to the second question.
Step 1: Calculate the area of the sector.
The area of a sector can be calculated using the formula:
Area = (θ/360°) × πr²
where θ is the central angle and r is the radius.
Given:
θ = 60°
r = 3 mi
Using the formula:
Area = (60/360) × π(3²)
Area = (1/6) × π × 9
Area = (π/6) × 9
Area = (3π/2) sq. mi
Therefore, the area of the sector is (3π/2) square miles.
To find an angle that is coterminal with -8π/3 between 0 and 2π, you can use the fact that coterminal angles will differ by a multiple of 2π.
First, convert -8π/3 into degrees. Recall that 2π radians is equivalent to 360 degrees.
-8π/3 * (180 degrees / π radians) = -480 degrees
Now, add or subtract a multiple of 360 degrees to find an angle between 0 and 2π.
-480 degrees + 360 degrees = -120 degrees
Therefore, an angle between 0 and 2π that is coterminal with -8π/3 is -120 degrees.