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The Six Trig Functions

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. The Six Trig Functions: Evaluate tan(5pi/4).

  1. Tangent is sine divided by cosine.
  2. Use the values at 5pi/4.
  3. tan(5pi/4) = 1.

Answer: 1

Practice problems

1. The Six Trig Functions: Give the unit-circle coordinates for 3pi/4.

Show solution
  1. Warm-up: First identify exactly what the question is asking: The Six Trig Functions: Give the unit-circle coordinates for 3pi/4.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Coordinates are (cos(theta), sin(theta)).
  4. At 3pi/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. The Six Trig Functions: Evaluate sin(5pi/6).

Show solution
  1. Warm-up: First identify exactly what the question is asking: The Six Trig Functions: Evaluate sin(5pi/6).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Sine is the y-coordinate.
  4. Use the point (-sqrt(3)/2, 1/2).
  5. sin(5pi/6) = 1/2.
  6. Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.

Answer: 1/2

3. The Six Trig Functions: Evaluate cos(7pi/6).

Show solution
  1. Warm-up: First identify exactly what the question is asking: The Six Trig Functions: Evaluate cos(7pi/6).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Cosine is the x-coordinate.
  4. Use the point (-sqrt(3)/2, -1/2).
  5. cos(7pi/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. The Six Trig Functions: Evaluate tan(5pi/4).

Show solution
  1. Core Practice: First identify exactly what the question is asking: The Six Trig Functions: Evaluate tan(5pi/4).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Tangent is sine divided by cosine.
  4. Use the values at 5pi/4.
  5. tan(5pi/4) = 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

5. The Six Trig Functions: If sin(theta) = -sqrt(3)/2, find csc(theta).

Show solution
  1. Core Practice: First identify exactly what the question is asking: The Six Trig Functions: If sin(theta) = -sqrt(3)/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) = -2sqrt(3)/3.
  6. Check the result by substituting or estimating: the response should match -2sqrt(3)/3 and make sense in the original problem.

Answer: -2sqrt(3)/3

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