To find the reference angle for the angle 19π/12, we first need to determine what quadrant this angle lies in and then calculate its reference angle accordingly.
The angle 19π/12 can be simplified as follows:
- 19π/12 is greater than 2π (which is equivalent to 24π/12), meaning it is an angle that has gone around the circle more than once, specifically 19 – 12 = 7π/12 into its second loop.
We can find the angle’s equivalent angle within the first full rotation by subtracting 2π from it:
- 19π/12 – 24π/12 = -5π/12
Since we want a positive coterminal angle, we can add 2π to bring it back into the positive range:
- -5π/12 + 24π/12 = 19π/12
Next, we divide 19π/12 by 2π to find how many full circles it has completed:
- 19π/12 divided by 2π = 19/(12*2) = 19/24, meaning this angle is 5π/12 into the second rotation.
Now, to find the reference angle, we take the angle’s position in relation to the nearest x-axis:
- In the second quadrant, the reference angle θ’ can be found as θ’ = π – θ. In this case:
- θ’ = π – (19π/12) = (12π/12 – 19π/12) = -7π/12.
- The positive reference angle would be: θ’ = 7π/12.
Thus, the reference angle for 19π/12 is:
- 7π/12 radians.