CMClearMathAcademy

Coordinates on the Unit Circle

A free Trigonometry lesson from the “The Unit Circle” unit, with a worked example and practice problems including step-by-step solutions.

On the unit circle, an angle's point has coordinates (cos(theta), sin(theta)). This is the bridge from triangles to graphs. In this lesson, the goal is to connect cosine to x-coordinates and sine to y-coordinates. Prerequisite check: Algebra II or College Algebra foundations. Example 1: at pi/3, the unit-circle point is (1/2, sqrt(3)/2), so cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2. Example 2: at 7pi/6, the reference angle is pi/6 and both sine and cosine are negative. A common mistake is memorizing a value without checking its sign; the safer habit is to find the reference angle, choose the special value, then apply the quadrant sign.

What you'll learn

Why it matters: The unit circle explains signals, waves, rotations, and why trig values repeat instead of being isolated facts.

Worked example

Problem. Example 1 Foundation: What does the unit-circle coordinate (cos(theta), sin(theta)) help you find?

  1. Cosine is x and sine is y.
  2. The unit circle turns trig values into coordinates.
  3. That connection powers graphing and equations.

Answer: sine and cosine values for the angle

Practice problems

1. Practice 1 Foundation: What unit-circle point matches pi/4?

Choices: (sqrt(2)/2, sqrt(2)/2) · (1, 0) · (0, 1) · (-1, 0)

Show solution
  1. Use the reference angle and quadrant.
  2. The point for pi/4 is (sqrt(2)/2, sqrt(2)/2).
  3. The x-coordinate is cosine and the y-coordinate is sine.

Answer: (sqrt(2)/2, sqrt(2)/2)

2. Practice 2 Setup: Find sin(2pi/3).

Choices: sqrt(3)/2 · -1/2 · -sqrt(3) · 1

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice 2 Setup: Find sin(2pi/3).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Sine is the y-coordinate.
  4. The point is (-1/2, sqrt(3)/2).
  5. So sin(2pi/3) = sqrt(3)/2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sqrt(3)/2

3. Practice 3 Meaning: Find cos(7pi/6).

Choices: -sqrt(3)/2 · -1/2 · sqrt(3)/3 · -1

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 3 Meaning: Find cos(7pi/6).
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. Cosine is the x-coordinate.
  4. The point is (-sqrt(3)/2, -1/2).
  5. So cos(7pi/6) = -sqrt(3)/2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: -sqrt(3)/2

4. Practice 4 Method: Where does 3pi/2 land?

Choices: negative y-axis · Quadrant I · Quadrant IV · positive x-axis

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 4 Method: Where does 3pi/2 land?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Locate the terminal side.
  4. 3pi/2 lands in negative y-axis.
  5. The location controls the signs.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: negative y-axis

5. Practice 5 Reasoning: Find tan(pi/6).

Choices: sqrt(3)/3 · 1/2 · sqrt(3)/2 · 0

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 5 Reasoning: Find tan(pi/6).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Tangent is sine divided by cosine.
  4. Using (sqrt(3)/2, 1/2), tan(pi/6) = sqrt(3)/3.
  5. If cosine is 0, tangent is undefined.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sqrt(3)/3

Practice this interactively with instant feedback and an AI tutor.

Practice Coordinates on the Unit Circle Take the free placement check

More Trigonometry lessons