The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system. $\theta = 30 ^ { \circ }$ , $( 1,3 )$
A) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 3 } { 2 } \right)$
B) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 1 } { 2 } \right)$
C) $\left( \frac { 3 + \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } - 1 } { 2 } \right)$
D) $\left( \frac { 3 + \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 1 } { 2 } \right)$
E) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } - 1 } { 2 } \right)$

Question

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system.​ $\theta = 30 ^ { \circ }$ , $( 1,3 )$ ​
A) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 3 } { 2 } \right)$
B) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 1 } { 2 } \right)$
C) $\left( \frac { 3 + \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } - 1 } { 2 } \right)$
D) $\left( \frac { 3 + \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } + 1 } { 2 } \right)$
E) $\left( \frac { 3 - \sqrt { 3 } } { 2 } , \frac { 3 \sqrt { 3 } - 1 } { 2 } \right)$

Quizplus · Accepted Answer

To find the coordinates of the point in the rotated coordinate system, we need to apply a counterclockwise rotation of 11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11 degrees to the point (11ea99db_b808_e677_94ac_41e75344b987_TB4590_11, 11ea99db_b809_0d88_94ac_0378418307b5_TB4590_11). Using the rotation formulas, the new coordinates are:

x' = x*cos(theta) - y*sin(theta)
y' = x*sin(theta) + y*cos(theta)

where theta is the rotation angle in radians, x and y are the original coordinates, and x' and y' are the new coordinates.

Plugging in the values, we get:

x' = 11ea99db_b808_e677_94ac_41e75344b987_TB4590_11*cos(11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11) - 11ea99db_b809_0d88_94ac_0378418307b5_TB4590_11*sin(11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11)
   = 11ea99db_b808_e677_94ac_41e75344b987_TB4590_11*cos(11ea99db_b7f7_405d_94ac_a9c530fc5f7b_TB4590_11) - 11ea99db_b809_0d88_94ac_0378418307b5_TB4590_11*sin(11ea99db_b7f7_405d_94ac_a9c530fc5f7b_TB4590_11)   (since 11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11 - 11ea99db_b7f7_405d_94ac_a9c530fc5f7b_TB4590_11 = 11ea99db_b808_9853_94ac_a54b488d990f_TB4590_11)
   = 11ea99db_b809_349b_94ac_ff88fa95d837_TB4590_11

y' = 11ea99db_b808_e677_94ac_41e75344b987_TB4590_11*sin(11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11) + 11ea99db_b809_0d88_94ac_0378418307b5_TB4590_11*cos(11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11)
   = 11ea99db_b808_e677_94ac_41e75344b987_TB4590_11*sin(11ea99db_b7f7_405d_94ac_a9c530fc5f7b_TB4590_11) + 11ea99db_b809_0d88_94ac_0378418307b5_TB4590_11*cos(11ea99db_b7f7_405d_94ac_a9c530fc5f7b_TB4590_11)   (since 11ea99db_b808_bf64_94ac_f73ea64d6422_TB4590_11 - 11ea99db_b7f7_405d_94ac_a9c530fc5f7b_TB4590_11 = 11ea99db_b808_9853_94ac_a54b488d990f_TB4590_11)
   = 11ea99db_b809_349a_94ac_e31531bc1a75_TB4590_11

Therefore, the answer is (C) 11ea99db_b809_349b_94ac_ff88fa95d837_TB4590_11.

The x′y′x ^ { \prime } y ^ { \prime }x′y′

The $x ^ { \prime } y ^ { \prime }$