For each angle or terminal ray described on the left, match it to the correct coordinates … — ACT

For each angle or terminal ray described on the left, match it to the correct coordinates $(x, y)$ of its intersection with the unit circle in the standard coordinate plane on the right.

The angle $\theta_1$ in standard position obtained by starting at the positive $x$ -axis, rotating counterclockwise by $\frac{13\pi}{3}$ radians, and then rotating clockwise by $570^\circ$ . $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$
The angle $\theta_2$ in standard position whose terminal ray passes through the point of intersection on the unit circle after a point starts at $(0, -1)$ and rotates counterclockwise by $\frac{23\pi}{4}$ radians. $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$
The angle $\theta_3$ in standard position that is coterminal with the angle $\phi = -1020^\circ$ . $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
The angle $\theta_4$ in standard position whose terminal ray is symmetric with respect to the $y$ -axis to the terminal ray of the angle $\frac{4\pi}{3}$ radians. $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$

Cevap

The correct matches associate the angle of 7pi/6 radians with coordinates (-sqrt(3)/2, -1/2); the angle of 5pi/4 radians with coordinates (-sqrt(2)/2, -sqrt(2)/2); the angle of pi/3 radians with coordinates (1/2, sqrt(3)/2); and the y-axis reflected angle of 4pi/3 radians with coordinates (1/2, -sqrt(3)/2).

Each description on the left simplifies to a unique angle in standard position on the unit circle. The coordinates on the right represent the exact

(\cos\theta, \sin\theta)

values for each corresponding angle: the angle of

\frac{7\pi}{6}

radians corresponds to coordinates

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

; the angle of

\frac{5\pi}{4}

radians corresponds to coordinates

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

; the angle of

\frac{\pi}{3}

radians corresponds to coordinates

\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

; and the terminal ray symmetric to the terminal ray of

\frac{4\pi}{3}

radians across the

y

-axis corresponds to coordinates

\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

Adım Adım Çözüm

Analyze the first angle description by converting all angle measures to a common unit (radians) and computing the net rotation.

For the first angle,

570^\circ = \frac{19\pi}{6}

radians, and the net rotation is

\frac{13\pi}{3} - \frac{19\pi}{6} = \frac{7\pi}{6}

radians.

Converting degrees to radians and subtracting the clockwise rotation from the counterclockwise rotation simplifies the angle to a standard radian value.

For the second angle, express the starting point as an angle in radians and add the given counterclockwise rotation, then find the coterminal angle in

[0, 2\pi)

Starting at

(0, -1)

is equivalent to

\frac{3\pi}{2}

radians. Total rotation is

\frac{3\pi}{2} + \frac{23\pi}{4} = \frac{29\pi}{4}

radians, which is coterminal with

\frac{29\pi}{4} - 6\pi = \frac{5\pi}{4}

radians.

Using the standard position angle of the starting point allows us to sum the rotations and determine the final terminal ray position.

Determine the coterminal angle for

-1020^\circ

in the interval

[0, 360^\circ)

and convert it to radians.

-1020^\circ + 1080^\circ = 60^\circ

, which equals

\frac{\pi}{3}

radians.

Adding multiples of

360^\circ

finds the equivalent positive angle within one full revolution.

Find the coordinates of the terminal ray of

\frac{4\pi}{3}

radians on the unit circle, reflect the point across the

y

-axis, and determine the coordinates of the resulting point.

The coordinate point of

\frac{4\pi}{3}

\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

. Reflecting this point across the

y

-axis yields

\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

Symmetry across the

y

-axis negates the

x

-coordinate of the point on the unit circle.

Match each of the simplified angles to their corresponding standard coordinates

(x, y) = (\cos\theta, \sin\theta)

on the unit circle.

The angle

\frac{7\pi}{6}

matches

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

;

\frac{5\pi}{4}

matches

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

;

\frac{\pi}{3}

matches

\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

; and the reflected terminal ray matches

\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

Evaluating the sine and cosine functions at each angle yields the final coordinates.

Anahtar Kavram

Identifying terminal coordinates of angles on the unit circle by converting between degrees and radians, calculating coterminal angles, and applying coordinate symmetries.

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