Questions: The Unit Circle

Question 1

What are the coordinates of the point on the unit circle corresponding to θ = 3π/4?

• A. (-√2/2, √2/2)
• B. (√2/2, √2/2)
• C. (-√2/2, -√2/2)
• D. (√2/2, -√2/2)

Answer: (A)

θ = 3π/4 is in the second quadrant (90° to 180°), where x is negative and y is positive. The reference angle is π/4, whose unit circle coordinates are (√2/2, √2/2). In quadrant II, x becomes negative: (-√2/2, √2/2). Options B and D are in quadrant I and IV respectively, and option C is in quadrant III.

Question 2

The coordinates of a point on the unit circle at angle θ are given by (sin θ, cos θ).

• T. True
• F. False

Answer: False

The coordinates are (cos θ, sin θ) — cosine is the x-coordinate and sine is the y-coordinate. This follows from the definitions: in a right triangle inscribed in the unit circle with hypotenuse 1, cos θ = adjacent/hypotenuse = x/1 = x, and sin θ = opposite/hypotenuse = y/1 = y. Swapping them is one of the most common errors when first learning the unit circle.

Question 3

Why does the unit circle allow sine and cosine to be defined for any real number, not just angles between 0° and 90°?

Model answer: Because any angle — no matter how large or what sign — traces out a point on the circle, and we simply define cos θ and sin θ as the x- and y-coordinates of that point. The right-triangle definition breaks down outside 0°–90°, but the coordinate definition works everywhere.

The right-triangle definition requires an acute angle inside a triangle, so it cannot handle angles like 150°, 270°, or -45°. By redefining cos θ and sin θ as the x and y coordinates of the point reached by rotating θ radians counterclockwise from (1, 0) on the unit circle, the definitions extend naturally to all real inputs — including multiple full rotations and negative angles (clockwise rotation).