College Trigonometry, 2011a

922 Applications of Trigonometry
To find alternate descriptions for P , we note that the distance from P tothepoleis4units,so
any representation (r, θ) forP must have r = ±4. As we noted above, P lies on the terminal
side of π 6 , so this, coupled with r =4,givesus( 4, π )
6 as one of our answers. To find a different
representation for P with r = −4, we may choose any angle coterminal with the angle in the
original representation of P ( −4, 7π )
6 .Wepick−
5π
6 and get ( −4, − 5π )
6 as our second answer.
P ( 4, π )
6
P ( −4, − 5π 6
)
θ = − 5π 6
θ = π 6
Pole
Pole
3. To plot P ( 117, − 5π )
2 , we move along the polar axis 117 units from the pole and rotate
clockwise 5π 2
radians as illustrated below.
Pole
Pole
θ = − 5π 2
P ( 117, − 5π 2
)
Since P is 117 units from the pole, any representation (r, θ) forP satisfies r = ±117. For the
r = 117 case, we can take θ to be any angle coterminal with − 5π 2
. In this case, we choose
θ = 3π 2 , and get ( 117, 3π )
2 as one answer. For the r = −117 case, we visualize moving left 117
units from the pole and then rotating through an angle θ to reach P . We find that θ = π 2
satisfies this requirement, so our second answer is ( −117, π )
2 .
Pole
Pole
θ = 3π 2
θ = π 2
P ( 117, 3π 2
)
P ( −117, π )
2