Question
A point on the unit circle is (1/2,3√2) . What is the sine of the angle formed by a terminal ray that intersects the unit circle at this point?(1 point)
Responses
12
Start Fraction 1 over 2 End Fraction
−12
negative Start Fraction 1 over 2 End Fraction
−3√2
negative Start Fraction Start Root 3 End Root over 2 End Fraction
3√2
Responses
12
Start Fraction 1 over 2 End Fraction
−12
negative Start Fraction 1 over 2 End Fraction
−3√2
negative Start Fraction Start Root 3 End Root over 2 End Fraction
3√2
Answers
GPT-4o mini
The coordinates given, \((\frac{1}{2}, 3\sqrt{2})\), do not represent a point on the unit circle, as the unit circle consists of points \((x, y)\) that satisfy the equation \(x^2 + y^2 = 1\).
To verify:
\[
\left( \frac{1}{2} \right)^2 + (3\sqrt{2})^2 = \frac{1}{4} + 18 = \frac{1}{4} + 72/4 = \frac{73}{4}
\]
Since \(\frac{73}{4} \neq 1\), this point is not on the unit circle.
However, since you've asked for the sine of the angle formed by a terminal ray that intersects the unit circle, if we want to find the sine of the angle for a point that is on the unit circle, we need to use its correct coordinates.
If we were to take the correct coordinates that are on the unit circle, we would use coordinates like \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) which would yield the correct answer.
If we found a point on the unit circle at \((x, y)\), the sine of the angle (the y-coordinate) would be:
Given that the point must actually be on the unit circle and we only found it as such above, please verify the correct coordinate on the unit circle or provide further information if necessary.
For standard angle corresponding with x value of \(\frac{1}{2}\), we find:
1. The x-coordinate corresponding to \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) indicates an angle of \(\frac{\pi}{3}\) (60 degrees), where the sine is \(\frac{\sqrt{3}}{2}\).
2. For angles in other quadrants, sine may vary in sign according to corresponding angles.
Please clarify if there's any confusion or if incorrect information got provided about the coordinates to discuss the sine value accurately.
To verify:
\[
\left( \frac{1}{2} \right)^2 + (3\sqrt{2})^2 = \frac{1}{4} + 18 = \frac{1}{4} + 72/4 = \frac{73}{4}
\]
Since \(\frac{73}{4} \neq 1\), this point is not on the unit circle.
However, since you've asked for the sine of the angle formed by a terminal ray that intersects the unit circle, if we want to find the sine of the angle for a point that is on the unit circle, we need to use its correct coordinates.
If we were to take the correct coordinates that are on the unit circle, we would use coordinates like \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) which would yield the correct answer.
If we found a point on the unit circle at \((x, y)\), the sine of the angle (the y-coordinate) would be:
Given that the point must actually be on the unit circle and we only found it as such above, please verify the correct coordinate on the unit circle or provide further information if necessary.
For standard angle corresponding with x value of \(\frac{1}{2}\), we find:
1. The x-coordinate corresponding to \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) indicates an angle of \(\frac{\pi}{3}\) (60 degrees), where the sine is \(\frac{\sqrt{3}}{2}\).
2. For angles in other quadrants, sine may vary in sign according to corresponding angles.
Please clarify if there's any confusion or if incorrect information got provided about the coordinates to discuss the sine value accurately.