The angle measures associated with which set of ordered pairs share the same reference angle?
$\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right) ,\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$
$\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right),\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
$\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right),\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) ,\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
Answer (2)
Calculate the reference angle for each ordered pair by using the arctangent function and considering the quadrant.
Compare the reference angles within each set of ordered pairs.
Identify the set where both ordered pairs have the same reference angle.
Conclude that the set ( − 2 1 , − 2 3 ) and ( 2 1 , 2 3 ) shares the same reference angle of 60 degrees. Option 3
Explanation
Problem Analysis The problem requires us to identify the set of ordered pairs that correspond to angles sharing the same reference angle. We are given four sets of ordered pairs, and we need to determine the reference angle for each ordered pair in each set. The ordered pairs represent points on the unit circle, where the x-coordinate is the cosine of the angle and the y-coordinate is the sine of the angle. The reference angle is the acute angle formed by the terminal side of the angle and the x-axis.
Method To find the reference angle, we can use the arctangent function (atan2) to find the angle in standard position and then calculate the reference angle based on the quadrant in which the angle lies. The reference angle is always between 0 and 90 degrees.
Calculations and Comparison Let's analyze each option:
Option 1: ( − 2 3 , − 2 1 ) and ( − 2 1 , − 2 3 ) The reference angle for ( − 2 3 , − 2 1 ) is 30 degrees. The reference angle for ( − 2 1 , − 2 3 ) is 60 degrees. These angles do not share the same reference angle.
Option 2: ( 2 1 , − 2 3 ) and ( − 2 3 , 2 1 ) The reference angle for ( 2 1 , − 2 3 ) is 60 degrees. The reference angle for ( − 2 3 , 2 1 ) is 30 degrees. These angles do not share the same reference angle.
Option 3: ( − 2 1 , − 2 3 ) and ( 2 1 , 2 3 ) The reference angle for ( − 2 1 , − 2 3 ) is 60 degrees. The reference angle for ( 2 1 , 2 3 ) is 60 degrees. These angles share the same reference angle.
Option 4: ( 2 3 , 2 1 ) and ( 2 1 , 2 3 ) The reference angle for ( 2 3 , 2 1 ) is 30 degrees. The reference angle for ( 2 1 , 2 3 ) is 60 degrees. These angles do not share the same reference angle.
Final Answer Based on the calculations, the set of ordered pairs that share the same reference angle is ( − 2 1 , − 2 3 ) and ( 2 1 , 2 3 ) , where both angles have a reference angle of 60 degrees.
Examples
Understanding reference angles is crucial in various fields, such as physics and engineering, when dealing with periodic phenomena like oscillations or waves. For instance, when analyzing the motion of a pendulum, the reference angle helps determine the pendulum's displacement and velocity at different points in its swing. Similarly, in electrical engineering, reference angles are used to analyze alternating current (AC) circuits and determine the phase relationships between voltage and current.
The ordered pairs \\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right) and \\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) share the same reference angle of 60 degrees. Therefore, the correct option is Option 3.
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