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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

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. How many degrees are in pi radians?

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  1. pi radians equals 180 degrees.

Answer: 180

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

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

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  1. Cosine gives x.

Answer: x-coordinate

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

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

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  1. 90 degrees is the top of the circle.

Answer: (0, 1)

4. What is sin(90 degrees)?

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  1. At 90 degrees, the y-coordinate is 1.

Answer: 1

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

Choices: Input · Output · Amplitude · Denominator

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  1. A function takes an input and returns an output.
  2. For sine, the angle is the input.
  3. So theta is the input.

Answer: Input

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