$$ \left. \begin{array} { l } { X + S imes \cos ( \frac { \pi imes heta } { 180 } ) } \ { Y + S imes \sin ( \frac { \pi imes heta } { 180 } ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { X + S 	imes \cos ( \frac { \pi 	imes 	heta } { 180 } ) } \ { Y + S 	imes \sin ( \frac { \pi 	imes 	heta } { 180 } ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

1. We start with a coordinate point $$(X, Y)$$ and a displacement of length $$S$$ in a direction given by an angle $$\theta$$ (measured in degrees).  
2. To calculate the new coordinates after moving $$S$$ units in direction $$\theta$$, we must convert the angle from degrees to radians because the trigonometric functions $$\cos$$ and $$\sin$$ use radian measure. The conversion is done as follows:  
   $$
   \theta_{\text{radians}} = \frac{\pi \times \theta}{180}
   $$
3. The displacement in the $$x$$-direction is given by:
   $$
   \Delta x = S \times \cos\left(\frac{\pi \times \theta}{180}\right)
   $$
4. Similarly, the displacement in the $$y$$-direction is:
   $$
   \Delta y = S \times \sin\left(\frac{\pi \times \theta}{180}\right)
   $$
5. Adding these displacements to the original coordinates yields the new position:
   $$
   \begin{cases}
   X_{\text{new}} = X + \Delta x = X + S \times \cos\left(\frac{\pi \times \theta}{180}\right) \
   Y_{\text{new}} = Y + \Delta y = Y + S \times \sin\left(\frac{\pi \times \theta}{180}\right)
   \end{cases}
   $$

Thus, the final coordinates are given by:
$$
\left.
\begin{array}{l}
X + S \times \cos\left(\frac{\pi \times \theta}{180}\right)\[6pt]
Y + S \times \sin\left(\frac{\pi \times \theta}{180}\right)
\end{array}
\right.
$$
The new coordinates after moving $$ S $$ units at an angle $$ \theta $$ from the original point $$(X, Y)$$ are: $$ \begin{cases} X_{\text{new}} = X + S \times \cos\left(\frac{\pi \times \theta}{180}\right) \ Y_{\text{new}} = Y + S \times \sin\left(\frac{\pi \times \theta}{180}\right) \end{cases} $$

\( \left. \begin{array} { l } { X + S \times \cos ( \frac { \pi \times \theta } { 180 } ) } \\ { Y + S \times \sin ( \frac { \pi \times \theta } { 180 } ) } \end{array} \right. \)

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