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An angle of measure theta is in standard position with initial ray on the x-axis, vertex at the origin, and terminal ray in Quadrant 2.The terminal ray in Quadrant 2 intersects the circle at point P.A line segment begins at the origin and ends at point R, which lies on the circle in Quadrant 3.A dashed vertical line segment extends from point P to point R. The segment is perpendicular to the x axis.A dashed horizontal line segment extends from point P to a point on the y axis. The segment is perpendicular to the y axis.​​​The figure shows a circle centered at the origin with an angle of measure 𝜃 radians in standard position. The terminal ray of the angle intersects the circle at point 𝑃, and point 𝑅 also lies on the circle. The coordinates of 𝑃 are (𝑥,𝑦), and the coordinates of 𝑅 are (𝑥,-𝑦). Which of the following is true about the sine of 𝜃 ?Responses​ sin⁡𝜃=𝑥5 , because it is the ratio of the horizontal displacement of 𝑃 from the 𝑦-axis to the distance between the origin and 𝑃.​ sin theta equals x over 5  , because it is the ratio of the horizontal displacement of  P  from the  y -axis to the distance between the origin and  P .​ sin⁡𝜃=𝑥5 , because it is the ratio of the horizontal displacement of 𝑅 from the 𝑦-axis to the distance between the origin and 𝑅.​ sin theta equals x over 5  , because it is the ratio of the horizontal displacement of  R  from the  y -axis to the distance between the origin and  R .​​​ sin⁡𝜃=-𝑦5 , because it is the ratio of the vertical displacement of 𝑅 from the 𝑥-axis to the distance between the origin and 𝑅.​​​ sin theta equals fraction numerator negative y over denominator 5 end fraction  , because it is the ratio of the vertical displacement of  R  from the  x -axis to the distance between the origin and  R .​​ sin⁡𝜃=𝑦5 , because it is the ratio of the vertical displacement of 𝑃 from the 𝑥-axis to the distance between the origin and 𝑃.

Question

An angle of measure theta is in standard position with initial ray on the x-axis, vertex at the origin, and terminal ray in Quadrant 2.The terminal ray in Quadrant 2 intersects the circle at point P.A line segment begins at the origin and ends at point R, which lies on the circle in Quadrant 3.A dashed vertical line segment extends from point P to point R. The segment is perpendicular to the x axis.A dashed horizontal line segment extends from point P to a point on the y axis. The segment is perpendicular to the y axis.​​​The figure shows a circle centered at the origin with an angle of measure 𝜃 radians in standard position. The terminal ray of the angle intersects the circle at point 𝑃, and point 𝑅 also lies on the circle. The coordinates of 𝑃 are (𝑥,𝑦), and the coordinates of 𝑅 are (𝑥,-𝑦). Which of the following is true about the sine of 𝜃 ?Responses​ sin⁡𝜃=𝑥5 , because it is the ratio of the horizontal displacement of 𝑃 from the 𝑦-axis to the distance between the origin and 𝑃.​ sin theta equals x over 5  , because it is the ratio of the horizontal displacement of  P  from the  y -axis to the distance between the origin and  P .​ sin⁡𝜃=𝑥5 , because it is the ratio of the horizontal displacement of 𝑅 from the 𝑦-axis to the distance between the origin and 𝑅.​ sin theta equals x over 5  , because it is the ratio of the horizontal displacement of  R  from the  y -axis to the distance between the origin and  R .​​​ sin⁡𝜃=-𝑦5 , because it is the ratio of the vertical displacement of 𝑅 from the 𝑥-axis to the distance between the origin and 𝑅.​​​ sin theta equals fraction numerator negative y over denominator 5 end fraction  , because it is the ratio of the vertical displacement of  R  from the  x -axis to the distance between the origin and  R .​​ sin⁡𝜃=𝑦5 , because it is the ratio of the vertical displacement of 𝑃 from the 𝑥-axis to the distance between the origin and 𝑃.

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Solution

The correct answer is "sin⁡𝜃=𝑦/5, because it is the ratio of the vertical displacement of 𝑃 from the 𝑥-axis to the distance between the origin and 𝑃."

In a unit circle, the sine of an angle in standard position is defined as the y-coordinate of the point where the terminal side of the angle intersects the circle. In this case, the y-coordinate of point P is given as y, and the radius (distance from the origin to P) is given as 5. Therefore, the sine of the angle is y/5.

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