What are the coordinates of the terminal point determined by [tex]$t=\frac{20 \pi}{3}$[/tex]?
A. [tex]$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$[/tex]
B. [tex]$\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$[/tex]
C. [tex]$\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$[/tex]
D. [tex]$\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$[/tex]
Answer (2)
Find the coterminal angle of 3 20 π within the interval [ 0 , 2 π ) , which is 3 2 π .
Evaluate the cosine and sine of the coterminal angle: cos ( 3 2 π ) = − 2 1 and sin ( 3 2 π ) = 2 3 .
Determine the coordinates of the terminal point as ( cos ( 3 2 π ) , sin ( 3 2 π )) .
State the final coordinates: ( − 2 1 , 2 3 ) .
Explanation
Problem Analysis We are asked to find the coordinates of the terminal point determined by t = 3 20 π . The coordinates of the terminal point for an angle t on the unit circle are given by ( cos t , sin t ) . Therefore, we need to find cos ( 3 20 π ) and sin ( 3 20 π ) .
Find Coterminal Angle First, we need to find a coterminal angle to 3 20 π that lies in the interval [ 0 , 2 π ) . To do this, we can divide 3 20 π by 2 π to find the number of full rotations and the remaining angle.
Determine Coterminal Angle 3 20 π = 3 20 π = 6 π + 3 2 π So, 3 20 π is coterminal with 3 2 π . This means that the terminal point for 3 20 π is the same as the terminal point for 3 2 π .
Evaluate Cosine and Sine Now we need to evaluate cos ( 3 2 π ) and sin ( 3 2 π ) . Recall that 3 2 π is in the second quadrant, where cosine is negative and sine is positive. The reference angle for 3 2 π is π − 3 2 π = 3 π .
Find Coordinates We know that cos ( 3 π ) = 2 1 and sin ( 3 π ) = 2 3 . Therefore, cos ( 3 2 π ) = − 2 1 sin ( 3 2 π ) = 2 3 Thus, the coordinates of the terminal point are ( − 2 1 , 2 3 ) .
Final Answer The coordinates of the terminal point determined by t = 3 20 π are ( − 2 1 , 2 3 ) .
Examples
Understanding terminal points on the unit circle is crucial in fields like physics and engineering. For instance, when analyzing simple harmonic motion, such as the oscillation of a pendulum or the movement of a spring, the position of the object can be described using trigonometric functions related to angles on the unit circle. If you're designing a mechanical system where components rotate, knowing the coordinates of specific angles helps predict the position and velocity of those components at any given time, ensuring smooth and efficient operation.
The coordinates of the terminal point determined by t = 3 20 π are ( − 2 1 , 2 3 ) , corresponding to option D. This is found by determining the coterminal angle, which is 3 2 π , and evaluating the sine and cosine for that angle.
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