Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What coordinate pair corresponds with the angle π/4?

1. (1/2, √3/2)

2. (√2/2, √2/2)

3. (√3/2, 1/2)

4. (0, 1)

The correct answer is: (√2/2, √2/2)

The angle \( \pi/4 \) radians, which is equivalent to 45 degrees, is notable for its position in the unit circle. When considering points on the unit circle, the coordinates of these points can be expressed as \( (x, y) = (\cos(\theta), \sin(\theta)) \) for a given angle \( \theta \). At \( \pi/4 \), both the cosine and sine are equal, taking on the value of \( \sqrt{2}/2 \). This is derived from the fact that a 45-degree angle creates an isosceles right triangle where the lengths of the legs are equal, maximizing the height and base within the confines of the unit circle's radius of 1. Thus, the coordinate pair corresponding to the angle \( \pi/4 \) on the unit circle is \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \), confirming that the correct response aligns perfectly with the standard properties of trigonometric functions at this specific angle. The other options do not yield the equal values necessary for the coordinates at \( \pi/4 \), further establishing the uniqueness of this solution.