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Special Angle Exact Values

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. Special Angle Exact Values: 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. Special Angle Exact Values: Give the unit-circle coordinates for 2pi/3.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Special Angle Exact Values: Give the unit-circle coordinates for 2pi/3.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  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. Special Angle Exact Values: Evaluate sin(3pi/4).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Special Angle Exact Values: Evaluate sin(3pi/4).
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  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. Special Angle Exact Values: Evaluate cos(5pi/6).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Special Angle Exact Values: Evaluate cos(5pi/6).
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  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. Special Angle Exact Values: Evaluate tan(7pi/6).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Special Angle Exact Values: Evaluate tan(7pi/6).
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  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. Special Angle Exact Values: If sin(theta) = -sqrt(2)/2, find csc(theta).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Special Angle Exact Values: 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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