A robotic arm rotates counterclockwise by 210^\circ and then rotates clockwise by 75^\circ… — SAT

A robotic arm rotates counterclockwise by $210^\circ$ and then rotates clockwise by $75^\circ$ . What is the net counterclockwise rotation of the robotic arm, in radians?

$\frac{3\pi}{4}$ Answer
B
$\frac{4\pi}{3}$
C
$\frac{29\pi}{36}$
D
$\frac{3\pi}{5}$

Answer

The correct answer is

\frac{3\pi}{4}

radians.

The correct answer is determined by first finding the net angle of rotation. Subtracting the

75^\circ

clockwise rotation from the

210^\circ

counterclockwise rotation yields a net rotation of

135^\circ

counterclockwise. Converting this angle to radians is done by multiplying

135^\circ

\frac{\pi}{180^\circ}

, resulting in

\frac{135\pi}{180}

radians. Dividing both the numerator and denominator by 45 simplifies the fraction to

\frac{3\pi}{4}

radians.

Step-by-Step Solution

Calculate the net counterclockwise rotation in degrees.

210^\circ - 75^\circ = 135^\circ

Since counterclockwise rotation is positive and clockwise rotation is negative, we subtract the clockwise rotation from the counterclockwise rotation.

Convert the angle from degrees to radians.

135 \times \frac{\pi}{180} = \frac{135\pi}{180}

radians

To convert degrees to radians, multiply the degree measure by the conversion factor

\frac{\pi}{180^\circ}

Simplify the fraction to its lowest terms.

\frac{3\pi}{4}

radians

Divide both the numerator and the denominator by their greatest common divisor, which is 45.

Key Concept

To convert an angle from degrees to radians, multiply the degree measure by

\frac{\pi}{180^\circ}

and simplify the resulting fraction.

Alternative Method

Alternatively, convert each individual rotation to radians first, then subtract. Convert

210^\circ

to radians:

210 \times \frac{\pi}{180} = \frac{7\pi}{6}

radians. Convert

75^\circ

to radians:

75 \times \frac{\pi}{180} = \frac{5\pi}{12}

radians. Subtract the clockwise rotation from the counterclockwise rotation:

\frac{7\pi}{6} - \frac{5\pi}{12} = \frac{14\pi}{12} - \frac{5\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4}

radians.

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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