Question Video: Exploring the Different Angles between 0 and 2𝜋 That Have the Same Trigonometric Function Mathematics

Consider 𝐴, a point on a unit circle corresponding to the angle of 3𝜋/2. Is there another point on the unit circle that has the same 𝑦-coordinate as 𝐴 and represents an angle in the interval [0, 2𝜋)? If yes, give the angle.

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Video Transcript

Consider 𝐴, a point on a unit circle corresponding to the angle of three 𝜋 over two. Is there another point on the unit circle that has the same 𝑦-coordinate as 𝐴 and represents an angle in the interval left-closed right-open from zero to two 𝜋? If yes, give the angle.

We will begin by sketching a unit circle as shown. We know that any angle in standard position is measured from the positive 𝑥-axis. If the angle is positive, as in this case, we measure in the counterclockwise direction. Since a full turn or rotation is equal to two 𝜋 radians, we can add on 𝜋 over two, 𝜋, three 𝜋 over two, and two 𝜋 radians to our diagram.

The terminal side of the angle three 𝜋 over two is the negative 𝑦-axis. This means that point 𝐴 has coordinates zero, negative one. This is the minimum point of the unit circle, and there is therefore no other point on the unit circle with 𝑦-coordinate equal to negative one. And we can therefore conclude that the correct answer is no. There is no other point that has the same 𝑦-coordinate as 𝐴 in the interval left-closed right-open from zero to two 𝜋.

If we completed a full rotation of two 𝜋 radians, we would have another point that has the same 𝑦-coordinate as 𝐴. However, this angle would not lie between zero and two 𝜋 radians.

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