What rotation about the origin is equivalent to [tex]R_{-200} \cdot[/tex] ?
A. [tex]R_{160}[/tex]
B. [tex]R_{200}[/tex]
C. [tex]R_{-160}[/tex]
D. [tex]R_{560}[/tex]
Answer (1)
Recognize that rotations are periodic with a period of 360 degrees.
Add 360 degrees to -200 degrees to find an equivalent angle: − 200 + 360 = 160 .
Verify that R 160 is indeed equivalent to R − 200 .
Conclude that the equivalent rotation is R 160 .
Explanation
Analyze the problem The problem asks us to find a rotation about the origin that is equivalent to a rotation of R − 200 . This means we need to find an angle that, when rotated about the origin, results in the same final position as a rotation of -200 degrees.
Use periodicity of rotations Since rotations are periodic with a period of 360 degrees, we can add or subtract multiples of 360 degrees to the given angle to find an equivalent angle. In other words, a rotation of θ degrees is equivalent to a rotation of θ + 360 n degrees, where n is an integer.
Calculate equivalent angle To find an equivalent angle to -200 degrees, we can add 360 degrees: − 200 + 360 = 160 Therefore, a rotation of -200 degrees is equivalent to a rotation of 160 degrees.
Verify the options Now, let's check the given options: A. R 160 : This is equivalent to R − 200 since − 200 + 360 = 160 .
B. R 200 : This is not equivalent to R − 200 .
C. R − 160 : This is not equivalent to R − 200 .
D. R 560 : We can subtract 360 from 560 to find an equivalent angle: 560 − 360 = 200 . So, R 560 is equivalent to R 200 , which is not equivalent to R − 200 .
State the final answer Therefore, the rotation about the origin that is equivalent to R − 200 is R 160 .
Examples
Imagine you're steering a ship. A course correction of -200 degrees might be hard to visualize directly. Instead, you could make an equivalent correction of 160 degrees, achieving the same result but perhaps in a more intuitive way. This principle applies to many areas, from robotics to computer graphics, where understanding equivalent rotations simplifies complex maneuvers and calculations.
