CMClearMathAcademy

Unit 3 Review and Quiz

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

This checkpoint reviews The Unit Circle with a mix of computation, interpretation, and transfer problems.

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 3 Review and Quiz: Evaluate tan(7pi/6).

  1. Tangent is sine divided by cosine.
  2. Use the values at 7pi/6.
  3. tan(7pi/6) = sqrt(3)/3.

Answer: sqrt(3)/3

Practice problems

1. Unit 3 Review and Quiz: Give the unit-circle coordinates for 2pi/3.

Show solution
  1. Checkpoint Review: First identify exactly what the question is asking: Unit 3 Review and Quiz: Give the unit-circle coordinates for 2pi/3.
  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 2pi/3, the point is (-1/2, sqrt(3)/2).
  5. Use exact values.
  6. Check the result by substituting or estimating: the response should match (-1/2, sqrt(3)/2) and make sense in the original problem.

Answer: (-1/2, sqrt(3)/2)

2. Unit 3 Review and Quiz: Evaluate sin(3pi/4).

Show solution
  1. Checkpoint Review: First identify exactly what the question is asking: Unit 3 Review and Quiz: Evaluate sin(3pi/4).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Sine is the y-coordinate.
  4. Use the point (-sqrt(2)/2, sqrt(2)/2).
  5. sin(3pi/4) = sqrt(2)/2.
  6. Check the result by substituting or estimating: the response should match sqrt(2)/2 and make sense in the original problem.

Answer: sqrt(2)/2

3. Unit 3 Review and Quiz: Evaluate cos(5pi/6).

Show solution
  1. Checkpoint Review: First identify exactly what the question is asking: Unit 3 Review and Quiz: Evaluate cos(5pi/6).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Cosine is the x-coordinate.
  4. Use the point (-sqrt(3)/2, 1/2).
  5. cos(5pi/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 3 Review and Quiz: Evaluate tan(7pi/6).

Show solution
  1. Checkpoint Review: First identify exactly what the question is asking: Unit 3 Review and Quiz: Evaluate tan(7pi/6).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Tangent is sine divided by cosine.
  4. Use the values at 7pi/6.
  5. tan(7pi/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 3 Review and Quiz: If sin(theta) = -sqrt(2)/2, find csc(theta).

Show solution
  1. Checkpoint Review: First identify exactly what the question is asking: Unit 3 Review and Quiz: 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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