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Unit Circle Basics

A free Algebra II lesson from the “Trigonometry and Modeling” unit, with a worked example and practice problems including step-by-step solutions.

The unit circle has radius 1. At an angle, cosine gives the x-coordinate and sine gives the y-coordinate. Radians measure angle by arc length on the unit circle.

What you'll learn

Why it matters: GPS heading calculations, mechanical-arm positions, and architectural angle layouts all reference the unit circle. The point (cos θ, sin θ) is the position at angle θ, and every other trig fact follows from that one picture.

Worked example

Problem. What are cos(0) and sin(0)?

  1. At 0 radians, the point on the unit circle is (1, 0).
  2. Cosine is the x-coordinate.
  3. Sine is the y-coordinate.

Answer: cos(0) = 1 and sin(0) = 0

Practice problems

1. How many degrees are in pi radians?

Show solution
  1. Warm-up: First identify exactly what the question is asking: How many degrees are in pi radians?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. pi radians equals 180 degrees.
  4. Check the result by substituting or estimating: the response should match 180 and make sense in the original problem.

Answer: 180

2. On the unit circle, cosine is the...

Choices: x-coordinate · y-coordinate · radius squared · slope only

Show solution
  1. Warm-up: First identify exactly what the question is asking: On the unit circle, cosine is the...
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Cosine gives x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x-coordinate

3. At 90 degrees, the unit-circle point is...

Choices: (0, 1) · (1, 0) · (0, -1) · (-1, 0)

Show solution
  1. Core Practice: First identify exactly what the question is asking: At 90 degrees, the unit-circle point is...
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. 90 degrees is the top of the circle.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (0, 1)

4. What is sin(90 degrees)?

Show solution
  1. Challenge: First identify exactly what the question is asking: What is sin(90 degrees)?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. At 90 degrees, the y-coordinate is 1.
  4. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

5. In sin(theta), theta is the...

Choices: Input · Output · Amplitude · Denominator

Show solution
  1. Mixed Review: First identify exactly what the question is asking: In sin(theta), theta is the...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A function takes an input and returns an output.
  4. For sine, the angle is the input.
  5. So theta is the input.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Input

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