Answer key: Unit Circle Worksheet

For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: Unit Circle Worksheet.

Goal: Read sine, cosine and tangent from the unit circle, connect degrees to radians, and use symmetry to predict values in every quadrant.

Connects to Common Core HSF-TF.A.2 (Explain how the unit circle extends trigonometric functions to all real numbers, interpreted as radian measures of angles.)

Completed results table

Values are the readouts of the tool (Unit circle) for each setting.

Answers

1. At 60°, the point on the circle is (cos θ, sin θ) = (½, √3/2). Show that sin²θ + cos²θ = 1 for this angle.

   (√3/2)² + (½)² = 3/4 + 1/4 = 1. With decimals: 0.866² + 0.5² ≈ 0.75 + 0.25 = 1. This works for every angle, because the point is always exactly 1 unit from the center (the Pythagorean theorem).

2. Why is tan 90° undefined?

   tan θ = sin θ ÷ cos θ. At 90° the point is (0, 1), so cos 90° = 0 and you cannot divide by zero.

3. Compare 30° and 150°. Which values are equal, which are opposite, and why?

   sin 30° = sin 150° = ½ = 0.5, but cos 150° = −√3/2 is the opposite of cos 30° = √3/2. The point for 150° is the mirror image of the 30° point across the y-axis: the height (sine) stays the same, and the x-coordinate (cosine) changes sign.

4. In which quadrant are both cos θ and sin θ negative? Give an angle from your table that shows it.

   Quadrant III (between 180° and 270°). At 225°, cos θ = sin θ = −√2/2 ≈ −0.7071, and tan θ = 1.

5. Convert 225° to radians by hand, then check your table.

   225 × π ÷ 180 = 5π/4 ≈ 3.927 radians.

6. Predict sin 330° and cos 330° without the tool, then check.

   330° is 30° below the positive x-axis, in Quadrant IV. So cos 330° = cos 30° = √3/2 ≈ 0.866, and sin 330° = −sin 30° = −½ = −0.5.