Part A
The given angle is 4pi/3. Multiply it by the factor 180/pi to convert from radians to degree mode.
Note how the given angle has pi in the numerator, while the conversion factor has pi in the denominator. The two pi terms will cancel.
(4pi/3)*(180/pi)
(4/3)*(180/1)
(4*180)/(3*1)
720/3
240
The angle 4pi/3 radians is equivalent to 240 degrees. This 240 degree angle is in quadrant 3. Any angle in this quadrant is between 180 degrees and 270 degrees, excluding both endpoints. This is the bottom left quadrant, aka the southwest quadrant.
Answer: Quadrant 3
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Part B
To find the reference angle, we'll subtract off pi. This only works for angles in quadrant 3. This is because the first pi radians, aka 180 degrees, is taken up by the first two upper quadrants. The remaining bit in the third quadrant is all we care about to find the reference angle.
reference angle = (given angle in quadrant 3) - pi
reference angle = (4pi/3) - pi
reference angle = (4pi/3) - (3pi/3)
reference angle = (4pi - 3pi)/3
reference angle = pi/3
Answer: pi/3 radians
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Part C
Use a calculator or a reference table to find that
tan(4pi/3) = tan(pi/3) = sqrt(3)
Alternatively, you can compute the sine and cosine values first
- sin(pi/3) = sqrt(3)/2
- cos(pi/3) = 1/2
Dividing the two items in the order mentioned will get us the tangent value
tan = sin/cos
tan(pi/3) = sin(pi/3) divide cos(pi/3)
tan(pi/3) = sqrt(3)/2 divide 1/2
tan(pi/3) = sqrt(3)
In the jump from the second to last step, to the last step, the denominators '2' cancel out when dividing.
Answer: sqrt(3)
