What rotation about the origin is equivalent to [tex]R_{-500^{\circ}}[/tex]?
A. [tex]R_{280}[/tex]
B. [tex]R_{-220}[/tex]
C. [tex]R_{220}[/tex]
D. [tex]R_{-280}[/tex]
Answer (2)
Find an equivalent angle by adding multiples of 36 0 ∘ to − 50 0 ∘ .
Add 36 0 ∘ to − 50 0 ∘ to get − 14 0 ∘ .
Add another 36 0 ∘ to − 14 0 ∘ to get 22 0 ∘ .
The equivalent rotation is R 220 .
Explanation
Understanding Equivalent Rotations We are given a rotation of − 50 0 ∘ about the origin, and we want to find an equivalent rotation. To do this, we can add or subtract multiples of 36 0 ∘ until we find an angle between 0 ∘ and 36 0 ∘ (or between − 36 0 ∘ and 0 ∘ ).
First Addition of 360 degrees Let's add 36 0 ∘ to − 50 0 ∘ : − 50 0 ∘ + 36 0 ∘ = − 14 0 ∘ This gives us an equivalent rotation of − 14 0 ∘ .
Second Addition of 360 degrees Now, let's add another 36 0 ∘ to − 14 0 ∘ : − 14 0 ∘ + 36 0 ∘ = 22 0 ∘ This gives us an equivalent rotation of 22 0 ∘ .
Finding the Correct Option So, a rotation of − 50 0 ∘ is equivalent to a rotation of 22 0 ∘ . Comparing this to the given choices, we see that option C, R 220 , is the correct answer.
Examples
Understanding rotations is crucial in many fields, such as computer graphics and robotics. For example, if you're programming a robot arm to rotate an object, you need to ensure the arm rotates to the correct final position, no matter how many full rotations it makes. Knowing how to find equivalent angles helps simplify the calculations and avoid errors in the robot's movements.
The rotation R − 50 0 ∘ is equivalent to the rotation R 22 0 ∘ after adding multiples of 36 0 ∘ to find an equivalent angle. The correct answer is Option C: R 220 .
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