To find the angle θ when sin θ = 1/3, locate the height 1/3 on the y‑axis of the unit circle and draw a horizontal auxiliary line at that height.[br]The points where this line intersects the unit circle correspond to the angles whose sine value is 1/3.[br]By measuring the angles from the positive x‑axis to these intersection points, you can determine the possible values of θ on the unit circle.

To find the angle θ when cos θ = −2/3, locate the value −2/3 on the x‑axis of the unit circle and draw a vertical auxiliary line at that position.[br]The points where this line intersects the unit circle correspond to the angles whose cosine value is −2/3.[br]By measuring the angles from the positive x‑axis to these intersection points, you can determine the possible values of θ on the unit circle.

To find the angle θ when tan θ = −2, you can use two equivalent visual approaches on the unit circle.[br]First, draw a line through the origin with slope −2; the points where this line intersects the unit circle represent the angles whose tangent value is −2.[br]Alternatively, locate the vertical line x = 1 and find the point on this line whose y‑coordinate is −2, which represents the ratio y/x = −2.[br]Drawing a line from the origin through this point will again intersect the unit circle at the same two positions.[br]By measuring the angles from the positive x‑axis to these intersection points, you can determine the possible values of θ on the unit circle.