Check all that apply.$\frac{\pi}{3}$ is the reference angle for:
A. $\frac{2 \pi}{3}$
B. $\frac{15 \pi}{3}$
C. $\frac{19 \pi}{3}$
D. $\frac{7 \pi}{3}$
Answer (1)
Determine the reference angle for 3 2 π : π − 3 2 π = 3 π .
Determine the reference angle for 3 15 π : 3 15 π = 5 π , which is coterminal with π , so the reference angle is 0.
Determine the reference angle for 3 19 π : 3 19 π − 6 π = 3 π , so the reference angle is 3 π .
Determine the reference angle for 3 7 π : 3 7 π − 2 π = 3 π , so the reference angle is 3 π .
The angles with a reference angle of 3 π are: A , C , D
Explanation
Understanding Reference Angles The reference angle is the acute angle formed by the terminal side of an angle and the x-axis. We are given that the reference angle is 3 π . We need to determine which of the given angles have this reference angle.
Checking Option A A. For the angle 3 2 π , it lies in the second quadrant. To find its reference angle, we subtract it from π : π − 3 2 π = 3 3 π − 3 2 π = 3 π So, 3 2 π has a reference angle of 3 π .
Checking Option B B. For the angle 3 15 π , we first simplify it: 3 15 π = 5 π Since 5 π = 2 ( 2 π ) + π , this angle is coterminal with π . The reference angle for π is 0, since it lies on the x-axis. So, 3 15 π does not have a reference angle of 3 π .
Checking Option C C. For the angle 3 19 π , we find a coterminal angle between 0 and 2 π by subtracting multiples of 2 π = 3 6 π : 3 19 π − 3 ( 2 π ) = 3 19 π − 3 18 π = 3 π Since the coterminal angle is 3 π , which is in the first quadrant, the reference angle is also 3 π .
Checking Option D D. For the angle 3 7 π , we find a coterminal angle between 0 and 2 π by subtracting 2 π = 3 6 π : 3 7 π − 3 6 π = 3 π Since the coterminal angle is 3 π , which is in the first quadrant, the reference angle is also 3 π .
Final Answer Therefore, the angles that have a reference angle of 3 π are 3 2 π , 3 19 π , and 3 7 π .
Examples
Reference angles are useful in trigonometry because trigonometric functions of an angle and its reference angle differ only in sign. For example, if you need to find the sine of 3 2 π , you can find the sine of its reference angle 3 π , which is 2 3 . Since 3 2 π is in the second quadrant, where sine is positive, sin ( 3 2 π ) = 2 3 . This simplifies calculations and helps in understanding trigonometric values for various angles.
