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Unit Circle Fluency

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

The unit circle records each angle as a coordinate pair (cos theta, sin theta). Exact values, the six trig functions, reciprocal relationships, and quadrant signs all come from that coordinate model.

What you'll learn

Why it matters: Circular motion, waves, direction, and rotation all use the unit circle to convert angles into coordinates.

Worked example

Problem. Unit Circle Fluency: Evaluate tan(pi/6).

  1. Tangent is sine divided by cosine.
  2. Use the values at pi/6.
  3. tan(pi/6) = sqrt(3)/3.

Answer: sqrt(3)/3

Practice problems

1. Unit Circle Fluency: Give the unit-circle coordinates for 5pi/4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Unit Circle Fluency: Give the unit-circle coordinates for 5pi/4.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Coordinates are (cos(theta), sin(theta)).
  4. At 5pi/4, the point is (-sqrt(2)/2, -sqrt(2)/2).
  5. Use exact values.
  6. Check the result by substituting or estimating: the response should match (-sqrt(2)/2, -sqrt(2)/2) and make sense in the original problem.

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

2. Unit Circle Fluency: Evaluate sin(4pi/3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Unit Circle Fluency: Evaluate sin(4pi/3).
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Sine is the y-coordinate.
  4. Use the point (-1/2, -sqrt(3)/2).
  5. sin(4pi/3) = -sqrt(3)/2.
  6. Check the result by substituting or estimating: the response should match -sqrt(3)/2 and make sense in the original problem.

Answer: -sqrt(3)/2

3. Unit Circle Fluency: Evaluate cos(11pi/6).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Unit Circle Fluency: Evaluate cos(11pi/6).
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Cosine is the x-coordinate.
  4. Use the point (sqrt(3)/2, -1/2).
  5. cos(11pi/6) = sqrt(3)/2.
  6. Check the result by substituting or estimating: the response should match sqrt(3)/2 and make sense in the original problem.

Answer: sqrt(3)/2

4. Unit Circle Fluency: Evaluate tan(pi/6).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Unit Circle Fluency: Evaluate tan(pi/6).
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Tangent is sine divided by cosine.
  4. Use the values at pi/6.
  5. tan(pi/6) = sqrt(3)/3.
  6. Check the result by substituting or estimating: the response should match sqrt(3)/3 and make sense in the original problem.

Answer: sqrt(3)/3

5. Unit Circle Fluency: If sin(theta) = sqrt(2)/2, find csc(theta).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Unit Circle Fluency: If sin(theta) = sqrt(2)/2, find csc(theta).
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. Cosecant is reciprocal sine.
  4. Take the reciprocal of sine.
  5. csc(theta) = sqrt(2).
  6. Check the result by substituting or estimating: the response should match sqrt(2) and make sense in the original problem.

Answer: sqrt(2)

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